Page 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1
Applying Measurement Uncertainty To Digital Multimeter Calibration
An introductory study of measurement
uncertainty and its application to digital
multimeter calibration
Teleconference
US amp Canada Toll Free Dial-In Number 1-(866) 230-5936
International Dial-In Number+1-281-913-1100
Conference Code 1010759559
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 2
Welcome
Greetings from ndash
Fluke Corporation
Everett Washington USA
We are very pleased to bring you this
presentation on measurement
uncertainty for DMM Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 3
Welcome
This presentation is based on Flukersquos
extensive experience with
minus Use and design of calibration
Instruments
minus Our experience and understanding of the
problems faced when applying
measurement uncertainty for both
regular and accredited metrology
Thanks for your time we hope you find it
both valuable and useful
Welcome and Thanks
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 4
Presented by
Flukersquos Calibration Business Unit
and Jack Somppi Electrical Calibration Instruments
Product Line Manager
jacksomppiflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 5
Web seminar etiquette
bull Choice of Audio ndash VOIP or Teleconference
minus VOIP receives audio only while teleconference is two way
sound
bull Donrsquot mute your phone if you have background
music enabled
bull Use QampA or chat to send me questions or request
clarification
bull There will be an opportunity throughout the
discussion to pause and ask questions
bull You can view the material using either full screen
or multi window methods
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6
Applying Measurement Uncertainty To Digital Multimeter Calibration
An introductory study of measurement
uncertainty and its application to digital
multimeter calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7
Objectives
In this session you will -
bull Be introduced to the concept of measurement
uncertainty and why it is important
bull Observe the basic elements that influence
measurement uncertainty for DMM calibration
applications
bull Study a simple but detailed example of calculating
measurement uncertainty
bull Consider some benefits of automating measurement
uncertainty calculations
bull Receive a variety of references for further research on
this topic
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8
Benefits
bull Introduce measurement uncertainty to
those in calibrationmetrology who are
not familiar with it
bull Understand why measurement
uncertainty is important for quality
metrology
bull Understand measurement uncertainty
with respect to DMM calibration
bull Appreciate to the benefits of automation
bull Have technical references for more
detailed information
bull Obtain copies of this presentation via
email
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9
Measurement Uncertainty amp Why It Is Important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10
Facts regarding measurement -
bull Can you ever measure the true value of
something
minus No there will always be errors
bull How important is this fact
minus Very important as measurement is never complete
unless you know how good it is
bull How is this taken into account in todayrsquos
calibration amp metrology
minus By applying amp documenting the measurement uncertainty
process to the tests being done
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11
Measurement uncertainty in metrology todayhellip
Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored
Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in
minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12
ISO 17025 ndash about measurement uncertaintyhellip
546 Estimation of uncertainty of
measurement
minus 5461 A calibration laboratory or a testing
laboratory performing its own calibrations shall
have and shall apply a procedure to estimate the
uncertainty of measurement for all calibrations
and types of calibrations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 2
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 2
Welcome
Greetings from ndash
Fluke Corporation
Everett Washington USA
We are very pleased to bring you this
presentation on measurement
uncertainty for DMM Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 3
Welcome
This presentation is based on Flukersquos
extensive experience with
minus Use and design of calibration
Instruments
minus Our experience and understanding of the
problems faced when applying
measurement uncertainty for both
regular and accredited metrology
Thanks for your time we hope you find it
both valuable and useful
Welcome and Thanks
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 4
Presented by
Flukersquos Calibration Business Unit
and Jack Somppi Electrical Calibration Instruments
Product Line Manager
jacksomppiflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 5
Web seminar etiquette
bull Choice of Audio ndash VOIP or Teleconference
minus VOIP receives audio only while teleconference is two way
sound
bull Donrsquot mute your phone if you have background
music enabled
bull Use QampA or chat to send me questions or request
clarification
bull There will be an opportunity throughout the
discussion to pause and ask questions
bull You can view the material using either full screen
or multi window methods
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6
Applying Measurement Uncertainty To Digital Multimeter Calibration
An introductory study of measurement
uncertainty and its application to digital
multimeter calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7
Objectives
In this session you will -
bull Be introduced to the concept of measurement
uncertainty and why it is important
bull Observe the basic elements that influence
measurement uncertainty for DMM calibration
applications
bull Study a simple but detailed example of calculating
measurement uncertainty
bull Consider some benefits of automating measurement
uncertainty calculations
bull Receive a variety of references for further research on
this topic
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8
Benefits
bull Introduce measurement uncertainty to
those in calibrationmetrology who are
not familiar with it
bull Understand why measurement
uncertainty is important for quality
metrology
bull Understand measurement uncertainty
with respect to DMM calibration
bull Appreciate to the benefits of automation
bull Have technical references for more
detailed information
bull Obtain copies of this presentation via
email
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9
Measurement Uncertainty amp Why It Is Important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10
Facts regarding measurement -
bull Can you ever measure the true value of
something
minus No there will always be errors
bull How important is this fact
minus Very important as measurement is never complete
unless you know how good it is
bull How is this taken into account in todayrsquos
calibration amp metrology
minus By applying amp documenting the measurement uncertainty
process to the tests being done
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11
Measurement uncertainty in metrology todayhellip
Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored
Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in
minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12
ISO 17025 ndash about measurement uncertaintyhellip
546 Estimation of uncertainty of
measurement
minus 5461 A calibration laboratory or a testing
laboratory performing its own calibrations shall
have and shall apply a procedure to estimate the
uncertainty of measurement for all calibrations
and types of calibrations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 3
Welcome
This presentation is based on Flukersquos
extensive experience with
minus Use and design of calibration
Instruments
minus Our experience and understanding of the
problems faced when applying
measurement uncertainty for both
regular and accredited metrology
Thanks for your time we hope you find it
both valuable and useful
Welcome and Thanks
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 4
Presented by
Flukersquos Calibration Business Unit
and Jack Somppi Electrical Calibration Instruments
Product Line Manager
jacksomppiflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 5
Web seminar etiquette
bull Choice of Audio ndash VOIP or Teleconference
minus VOIP receives audio only while teleconference is two way
sound
bull Donrsquot mute your phone if you have background
music enabled
bull Use QampA or chat to send me questions or request
clarification
bull There will be an opportunity throughout the
discussion to pause and ask questions
bull You can view the material using either full screen
or multi window methods
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6
Applying Measurement Uncertainty To Digital Multimeter Calibration
An introductory study of measurement
uncertainty and its application to digital
multimeter calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7
Objectives
In this session you will -
bull Be introduced to the concept of measurement
uncertainty and why it is important
bull Observe the basic elements that influence
measurement uncertainty for DMM calibration
applications
bull Study a simple but detailed example of calculating
measurement uncertainty
bull Consider some benefits of automating measurement
uncertainty calculations
bull Receive a variety of references for further research on
this topic
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8
Benefits
bull Introduce measurement uncertainty to
those in calibrationmetrology who are
not familiar with it
bull Understand why measurement
uncertainty is important for quality
metrology
bull Understand measurement uncertainty
with respect to DMM calibration
bull Appreciate to the benefits of automation
bull Have technical references for more
detailed information
bull Obtain copies of this presentation via
email
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9
Measurement Uncertainty amp Why It Is Important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10
Facts regarding measurement -
bull Can you ever measure the true value of
something
minus No there will always be errors
bull How important is this fact
minus Very important as measurement is never complete
unless you know how good it is
bull How is this taken into account in todayrsquos
calibration amp metrology
minus By applying amp documenting the measurement uncertainty
process to the tests being done
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11
Measurement uncertainty in metrology todayhellip
Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored
Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in
minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12
ISO 17025 ndash about measurement uncertaintyhellip
546 Estimation of uncertainty of
measurement
minus 5461 A calibration laboratory or a testing
laboratory performing its own calibrations shall
have and shall apply a procedure to estimate the
uncertainty of measurement for all calibrations
and types of calibrations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 4
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 4
Presented by
Flukersquos Calibration Business Unit
and Jack Somppi Electrical Calibration Instruments
Product Line Manager
jacksomppiflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 5
Web seminar etiquette
bull Choice of Audio ndash VOIP or Teleconference
minus VOIP receives audio only while teleconference is two way
sound
bull Donrsquot mute your phone if you have background
music enabled
bull Use QampA or chat to send me questions or request
clarification
bull There will be an opportunity throughout the
discussion to pause and ask questions
bull You can view the material using either full screen
or multi window methods
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6
Applying Measurement Uncertainty To Digital Multimeter Calibration
An introductory study of measurement
uncertainty and its application to digital
multimeter calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7
Objectives
In this session you will -
bull Be introduced to the concept of measurement
uncertainty and why it is important
bull Observe the basic elements that influence
measurement uncertainty for DMM calibration
applications
bull Study a simple but detailed example of calculating
measurement uncertainty
bull Consider some benefits of automating measurement
uncertainty calculations
bull Receive a variety of references for further research on
this topic
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8
Benefits
bull Introduce measurement uncertainty to
those in calibrationmetrology who are
not familiar with it
bull Understand why measurement
uncertainty is important for quality
metrology
bull Understand measurement uncertainty
with respect to DMM calibration
bull Appreciate to the benefits of automation
bull Have technical references for more
detailed information
bull Obtain copies of this presentation via
email
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9
Measurement Uncertainty amp Why It Is Important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10
Facts regarding measurement -
bull Can you ever measure the true value of
something
minus No there will always be errors
bull How important is this fact
minus Very important as measurement is never complete
unless you know how good it is
bull How is this taken into account in todayrsquos
calibration amp metrology
minus By applying amp documenting the measurement uncertainty
process to the tests being done
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11
Measurement uncertainty in metrology todayhellip
Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored
Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in
minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12
ISO 17025 ndash about measurement uncertaintyhellip
546 Estimation of uncertainty of
measurement
minus 5461 A calibration laboratory or a testing
laboratory performing its own calibrations shall
have and shall apply a procedure to estimate the
uncertainty of measurement for all calibrations
and types of calibrations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 5
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 5
Web seminar etiquette
bull Choice of Audio ndash VOIP or Teleconference
minus VOIP receives audio only while teleconference is two way
sound
bull Donrsquot mute your phone if you have background
music enabled
bull Use QampA or chat to send me questions or request
clarification
bull There will be an opportunity throughout the
discussion to pause and ask questions
bull You can view the material using either full screen
or multi window methods
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6
Applying Measurement Uncertainty To Digital Multimeter Calibration
An introductory study of measurement
uncertainty and its application to digital
multimeter calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7
Objectives
In this session you will -
bull Be introduced to the concept of measurement
uncertainty and why it is important
bull Observe the basic elements that influence
measurement uncertainty for DMM calibration
applications
bull Study a simple but detailed example of calculating
measurement uncertainty
bull Consider some benefits of automating measurement
uncertainty calculations
bull Receive a variety of references for further research on
this topic
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8
Benefits
bull Introduce measurement uncertainty to
those in calibrationmetrology who are
not familiar with it
bull Understand why measurement
uncertainty is important for quality
metrology
bull Understand measurement uncertainty
with respect to DMM calibration
bull Appreciate to the benefits of automation
bull Have technical references for more
detailed information
bull Obtain copies of this presentation via
email
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9
Measurement Uncertainty amp Why It Is Important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10
Facts regarding measurement -
bull Can you ever measure the true value of
something
minus No there will always be errors
bull How important is this fact
minus Very important as measurement is never complete
unless you know how good it is
bull How is this taken into account in todayrsquos
calibration amp metrology
minus By applying amp documenting the measurement uncertainty
process to the tests being done
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11
Measurement uncertainty in metrology todayhellip
Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored
Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in
minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12
ISO 17025 ndash about measurement uncertaintyhellip
546 Estimation of uncertainty of
measurement
minus 5461 A calibration laboratory or a testing
laboratory performing its own calibrations shall
have and shall apply a procedure to estimate the
uncertainty of measurement for all calibrations
and types of calibrations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 6
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6
Applying Measurement Uncertainty To Digital Multimeter Calibration
An introductory study of measurement
uncertainty and its application to digital
multimeter calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7
Objectives
In this session you will -
bull Be introduced to the concept of measurement
uncertainty and why it is important
bull Observe the basic elements that influence
measurement uncertainty for DMM calibration
applications
bull Study a simple but detailed example of calculating
measurement uncertainty
bull Consider some benefits of automating measurement
uncertainty calculations
bull Receive a variety of references for further research on
this topic
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8
Benefits
bull Introduce measurement uncertainty to
those in calibrationmetrology who are
not familiar with it
bull Understand why measurement
uncertainty is important for quality
metrology
bull Understand measurement uncertainty
with respect to DMM calibration
bull Appreciate to the benefits of automation
bull Have technical references for more
detailed information
bull Obtain copies of this presentation via
email
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9
Measurement Uncertainty amp Why It Is Important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10
Facts regarding measurement -
bull Can you ever measure the true value of
something
minus No there will always be errors
bull How important is this fact
minus Very important as measurement is never complete
unless you know how good it is
bull How is this taken into account in todayrsquos
calibration amp metrology
minus By applying amp documenting the measurement uncertainty
process to the tests being done
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11
Measurement uncertainty in metrology todayhellip
Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored
Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in
minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12
ISO 17025 ndash about measurement uncertaintyhellip
546 Estimation of uncertainty of
measurement
minus 5461 A calibration laboratory or a testing
laboratory performing its own calibrations shall
have and shall apply a procedure to estimate the
uncertainty of measurement for all calibrations
and types of calibrations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 7
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7
Objectives
In this session you will -
bull Be introduced to the concept of measurement
uncertainty and why it is important
bull Observe the basic elements that influence
measurement uncertainty for DMM calibration
applications
bull Study a simple but detailed example of calculating
measurement uncertainty
bull Consider some benefits of automating measurement
uncertainty calculations
bull Receive a variety of references for further research on
this topic
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8
Benefits
bull Introduce measurement uncertainty to
those in calibrationmetrology who are
not familiar with it
bull Understand why measurement
uncertainty is important for quality
metrology
bull Understand measurement uncertainty
with respect to DMM calibration
bull Appreciate to the benefits of automation
bull Have technical references for more
detailed information
bull Obtain copies of this presentation via
email
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9
Measurement Uncertainty amp Why It Is Important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10
Facts regarding measurement -
bull Can you ever measure the true value of
something
minus No there will always be errors
bull How important is this fact
minus Very important as measurement is never complete
unless you know how good it is
bull How is this taken into account in todayrsquos
calibration amp metrology
minus By applying amp documenting the measurement uncertainty
process to the tests being done
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11
Measurement uncertainty in metrology todayhellip
Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored
Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in
minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12
ISO 17025 ndash about measurement uncertaintyhellip
546 Estimation of uncertainty of
measurement
minus 5461 A calibration laboratory or a testing
laboratory performing its own calibrations shall
have and shall apply a procedure to estimate the
uncertainty of measurement for all calibrations
and types of calibrations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 8
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8
Benefits
bull Introduce measurement uncertainty to
those in calibrationmetrology who are
not familiar with it
bull Understand why measurement
uncertainty is important for quality
metrology
bull Understand measurement uncertainty
with respect to DMM calibration
bull Appreciate to the benefits of automation
bull Have technical references for more
detailed information
bull Obtain copies of this presentation via
email
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9
Measurement Uncertainty amp Why It Is Important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10
Facts regarding measurement -
bull Can you ever measure the true value of
something
minus No there will always be errors
bull How important is this fact
minus Very important as measurement is never complete
unless you know how good it is
bull How is this taken into account in todayrsquos
calibration amp metrology
minus By applying amp documenting the measurement uncertainty
process to the tests being done
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11
Measurement uncertainty in metrology todayhellip
Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored
Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in
minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12
ISO 17025 ndash about measurement uncertaintyhellip
546 Estimation of uncertainty of
measurement
minus 5461 A calibration laboratory or a testing
laboratory performing its own calibrations shall
have and shall apply a procedure to estimate the
uncertainty of measurement for all calibrations
and types of calibrations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 9
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9
Measurement Uncertainty amp Why It Is Important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10
Facts regarding measurement -
bull Can you ever measure the true value of
something
minus No there will always be errors
bull How important is this fact
minus Very important as measurement is never complete
unless you know how good it is
bull How is this taken into account in todayrsquos
calibration amp metrology
minus By applying amp documenting the measurement uncertainty
process to the tests being done
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11
Measurement uncertainty in metrology todayhellip
Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored
Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in
minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12
ISO 17025 ndash about measurement uncertaintyhellip
546 Estimation of uncertainty of
measurement
minus 5461 A calibration laboratory or a testing
laboratory performing its own calibrations shall
have and shall apply a procedure to estimate the
uncertainty of measurement for all calibrations
and types of calibrations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 10
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10
Facts regarding measurement -
bull Can you ever measure the true value of
something
minus No there will always be errors
bull How important is this fact
minus Very important as measurement is never complete
unless you know how good it is
bull How is this taken into account in todayrsquos
calibration amp metrology
minus By applying amp documenting the measurement uncertainty
process to the tests being done
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11
Measurement uncertainty in metrology todayhellip
Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored
Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in
minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12
ISO 17025 ndash about measurement uncertaintyhellip
546 Estimation of uncertainty of
measurement
minus 5461 A calibration laboratory or a testing
laboratory performing its own calibrations shall
have and shall apply a procedure to estimate the
uncertainty of measurement for all calibrations
and types of calibrations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 11
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11
Measurement uncertainty in metrology todayhellip
Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored
Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in
minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12
ISO 17025 ndash about measurement uncertaintyhellip
546 Estimation of uncertainty of
measurement
minus 5461 A calibration laboratory or a testing
laboratory performing its own calibrations shall
have and shall apply a procedure to estimate the
uncertainty of measurement for all calibrations
and types of calibrations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 12
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12
ISO 17025 ndash about measurement uncertaintyhellip
546 Estimation of uncertainty of
measurement
minus 5461 A calibration laboratory or a testing
laboratory performing its own calibrations shall
have and shall apply a procedure to estimate the
uncertainty of measurement for all calibrations
and types of calibrations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13
hellip about the sources of uncertaintyhellip
ISO 17025 Section 5463
minus NOTE 1 Sources contributing to the uncertainty
include but are not necessarily limited to
bull The reference standards and reference
materials used
bull Methods and equipment used
bull Environmental conditions
bull Properties and condition of the item being
tested or calibrated
bull Operator
There are many contributors to uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 14
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14
ISO 17025 Section 5104
Calibration Certificates shall include hellip
for the interpretation of calibration results
a The conditions of the test
b The uncertainty of measurement amp
compliance statements to metrological standards
c Evidence of traceability
When statements of compliance are made the
uncertainty of measurement shall be taken into account
hellipabout calibration certificateshellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 15
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15
An example of an accredited calibration certificate ndash
ldquoMeasurement uncertainties at the
time of test are given in the following
pages where applicable They are
calculated in accordance with the
method described in NIST TN1297
for a confidence level of 95 using a
coverage factor of approximately 2
(K=2)rdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 16
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16
To summarize the importance of measurement uncertaintyhellip
From the NPL UK - ldquoA Beginners Guide to
Uncertainty of Measurementrdquo
bull Uncertainty of a measurement tells us something about
its quality
bull Uncertainty of measurement is the doubt that exists
about the results of any measurement
bull For every measurement ndash even the most careful ndash there
is always a margin of doubt
bull You need to know the uncertainty before you can
decide whether the tolerance is met
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 17
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17
ldquoHow is this Measurement Uncertainty obtainedrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 18
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18
Properly Calculating Measurement Uncertainty ndash a topic often discussed amp
debated among metrologists
Initially there were no standardized
process to quantify measurement
uncertaintyhellip
But a standard technique was agreed
upon amp published in October 1993
ISO Guide 98 - Guide to the
Expression of Uncertainty in
Measurement (aka GUM)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 19
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19
In the USA refer to one of the Guides
relating to expressing of Uncertainty in
Measurement
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Recommendation Refer to the GUMs -
Internationally many metrology
organizations publish similar GUMs
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 20
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20
Questions - about measurement uncertainty or why it is important
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 21
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21
Measurement Uncertainty amp Calibrating DMMs
A study of applying the GUM to DMM
calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 22
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22
First ndash lets look at the concept
Our initial look ndash
bull Consider verifying a
precision digital multimeter
bull With a hypothetical study
of verifying the DMMrsquos
measurement performance
at 100 millivolts DC
bull Letrsquos briefly look at what
measurement uncertainty
could be in this case
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 23
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23
Some sources of measurement ldquodoubtrdquo when verifying a DMM
bull The most obvious amp significant sources of doubt
minus Inaccuracy of the calibratorrsquos output value
bull 1000000 mV might actually be 1000000 mV 0030 mV
minus Repeatability or randomness in measurement values from the DMM
bull 1000003 mV 999995 mV 1000010 mV etc
minus Resolution or sensitivity limits on the DMM
bull Itrsquos value is frac12 the least significant digit
bull in this example it represents 005 V
bull Many other factors that could also contribute to uncertainty
minus ambient temperature effects thermal emfs noise loading power line
conditions etc
bull Consider all factors and include if they significantly contribute to
measurement uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 24
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24
The GUMs classify two types of measurement uncertainty
bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)
minus For example Repeatability of the measurement (influenced by dmm
characteristics signal stability jitter noise etc)
bull Type B uncertainties ndash estimates of errors influencing the
measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)
minus Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)
minus Inherent limitations of the unit being tested (DMM resolution
limitations)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 25
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25
bull To quantify uncertainty the various sources of uncertainty need
to be quantified evaluated amp combined
bull Calculate a combined estimate of all the individual A and B types
of uncertainties
bull This combined uncertainty is
minus a basic estimate (representing one statistical standard deviation)
minus usually the RSS of all individual uncertainties
(Combining uncertainties using such an RSS technique applies to
uncertainties with standard relationships and are independent)
22
3
2
2
2
1 nc uuuuu
Combining all the uncertainties
cu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 26
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26
The expanded uncertainty
bull As mentioned calculations for uc pertain to plusmn one standard
deviation of measurement uncertainties (covering 68 of the population of measurements)
bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99
bull Expanding the calculated uncertainty through scaling estimates an
uncertainty that covers this larger population - Um
bull A coverage factor k (often equal to 2) would indicate a 95 confidence
ckuUm
68
95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 27
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27
Now returning to the hellip statement of uncertainty
bull A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate For example
VDMM = 1000051mV 00004 mV
bull In this case 00004 mV would be the resulting value
of Um calculated as shown below
ckumV Um00040
22
3
2
2
2
1 nuuuuk
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 28
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28
That describes the general process ndash are we okay so far
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 29
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29
Next a different and more detailed examplehellip
Examine the use of a Fluke 5500A to verify a 35 digit
DMM at 10 Amps of Alternating Current at 50 Hz
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 30
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30
bull Type A uncertainty is determined by the statistical
analysis of a series of observations (measurements)
bull Type A uncertainties includes effects from
minus Variations of multiple repeated readings from the UUT
minus Effects of the system noise
minus Noise and short term variation of the standard
bull Now letrsquos examine the basic statistics hellip
The ldquoArdquo portionhellip
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 31
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31
Measurement Value
1 1007
2 1002
3 1001
4 1006
5 1004
Average 1004
Measured value the average of a series of measurements
AIavg 0410
bull An average of multiple measurements is
a better estimate of the true value than
any individual value
bull As a rule of thumb taking between 4 amp
10 measurements are sufficient
bull Uncertainty improvements for more than
10 have diminishing results
bull In our example 5 readings are
sufficient Any improved uncertainties
for more readings are not significant
versus required measurement
tolerances (a typical DMM specification
for this example test is ~ plusmn25)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 32
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
Calculating the uncertainty due to measurement repeatability
bull The uncertainty is statistically
analyzed from the measurement
data series
u1 ndash for a normally distributed
population the best estimate of
uncertainty is the experimental
standard deviation of the mean NOTE In the unusual case where
1 the calibrating standard is extremely accurate amp
stable and
2 the repeated test measurement values are
unchanged (or even with only a plusmn one digit
change)
Then this uncertainty can be considered as non
significant
bull One measurement value would be sufficient
bull The type B resolution uncertainty is adequate
Experimental
Standard
Deviation
Experimental
Standard Deviation
of the Mean
1
)(1
2
n
xxn
i
i
s
n
su 1
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 33
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33
Measurement Value Deviation from
Average
x1 1007 +003
x2 1002 -002
x3 1001 -003
x4 1006 +002
x5 1004 000
x (Average) 1004
s (Estimated Std Dev) 002549
The estimated standard deviation
11
2
)(
n
i
n
i
xxs 255 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 34
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
Plus there are some other important characteristics to
consider
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
What are these
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 35
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35
Statistical terms amp concepts
bull Probability Distribution ldquothe scatter of the valuesrdquo
minus Normal or Gaussian
minus Rectangular or Uniform
minus Triangular U or bi-modal hellip
bull Degrees of Freedom ldquohow manyrdquo
minus A value related to the amount of information that was employed in
making the estimate
minus Usually equals the sample size minus one (n-1) for type A uncertainties
and is often considered infinite ( ) for parameters such as
manufacturer specifications
bull Sensitivity Coefficient ldquohow influentialrdquo
minus Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)
For more information see technical references on statistics
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 36
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36
u1 ndash estimated standard uncertainty
Calculate the Standard Deviation of the Mean
minus Probability Distribution = Normal
minus Sensitivity Coefficient = 1
minus Degrees of Freedom = 4
mAmAs
nu 411
5255
1
Grouped around a value
Direct influence on response
Based on 5 independent
measurements
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 37
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37
The rdquoBrdquo type of uncertainties hellip
All the other uncertainties that cannot be determined statistically during
the measurement process such as -
minus Calibrator inaccuracy or error
minus Measurement errors due to limitations of the DMMrsquos resolution
minus lead effects thermal emfs loading etc
bull Estimates here are based on scientific judgment using all relevant
information
bull Numerically these are expressed as one standard deviation
estimates for each different uncertainty
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 38
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38
u2 - uncertainty due to the calibrator inaccuracy
u2 is the plusmn1 sigma estimate of the calibrator error
bull (estimates a plusmn1 standard deviation coverage of
the errors - for 68 of all possible values)
bull based on the specifications for performance at the
specific test setting
minus Start with the manufacturerrsquos recommended specifications
at the test point
minus Adjust as required for any appropriate factors such as
legal traceability limitations improvements for output
characterizations etc
minus Convert to a plusmn one sigma confidence interval basis
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 39
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39
Refer to the calibrator specifications
bull For this example assume it is a certified calibrator that is routinely
calibrated every year
bull The absolute uncertainty specifications for 10 Amps 50 Hz
006 of output plus 2000 Amps
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 40
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40
Calculating u2
bull Step 1 Calculate the maximum instrument error per
manufacturerrsquos specifications at the point of test
5500A ndash 1 year specs 10 A 50 Hz
plusmn(006 of 10 A + 2000 μA)
is calculated to be
plusmn(6 mA + 2 mA) = plusmn8 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 41
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41
Calculating u2
bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)
minus If no other information is provided by the manufacturer assume a rectangular distribution
plusmn1σ = plusmnspec (radic3)
minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate
For example with a normal distribution at 99
plusmn1σ = plusmnspec (258)
Normal Probability Distribution
1 2 3-123
Uniform or Rectangular
Probability Distribution
Pro
ba
bili
ty o
f O
ccu
rre
nce
Value of Reading
Full width
Mean or
Average reading
-a +a
plusmnspec
limits
plusmnspec limits
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 42
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42
Flukersquos 5500A specifications
The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 43
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43
Calculating u2
bullThe value of u2 is the plusmn1 sigma calibrator spec
5500A ndash 1 year specs 10 A 50 Hz
This u2 value should be smaller than the published spec
With a spec of plusmn8 mA at 99 confidence
divide by 258 to convert to a plusmn1 sigma spec
u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 44
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44
Summary of u2 ndash
u2 is the plusmn1 sigma estimate of calibrator
specification uncertainty
minus Probability Distribution = Normal ndash as stated in the
manufacturerrsquos information
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu2 13
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 45
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45
u3 - uncertainty due to UUT measurement limitations
bull Measurements include error due to resolution limits of the UUT -
considered as one half of the LSD
bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA
1000 1000000
LSD (least significant digit)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 46
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46
Calculating u3
The formula for u3 is
Calculates the standard uncertainty related to one LSD
With an LSD of 10 mA -
u3 = 29 mA at a plusmn1 std dev
3LSD2
1 3 u
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 47
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47
Summary of u3 ndash
u3 is the plusmn1 sigma estimate of dmm LSD resolution
uncertainty
minus Probability Distribution = Rectangular
minus Sensitivity Coefficient = 1
minus Degrees of Freedom =
mAu3 92
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 48
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48
This completes the ldquoBrdquo portionhellip
u2 = 31 mA at plusmn1 standard deviation
u3 = 29 mA at plusmn1 standard deviation
bull There are no other ldquoBrdquo uncertainties which are
significant for this particular test (Note It is often good to identify and document the
other possible uncertainties deemed insignificant)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 49
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49
Combining all uncertainties hellip
A One Standard Deviation Estimate Of
Combined Uncertainty
Standard
Combined
Uncertainty
22
3
2
2
2
1 ncuuuuu
1216 mA 222 9213411
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 50
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50
Overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 810-3 1 Normal 258 3110-3
Resolution B u3 510-3 1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 121610-3 52
How do you calculate the overall
effective Degrees of Freedom
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 51
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51
Welch-Satterthwaite formula
bull is the overall effective
degrees of freedom for the
combined uncertainty (uc)
bull The formula considers each
uncertainty each sensitivity
coefficient and each
uncertaintyrsquos specific value
for degrees of freedom to
calculate
N
i i
ii
c
eff
v
xuc
yuv
1
44
4
)(
)(
veff
veff
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 52
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52
Welch-Satterthwaite formula in our example case
25)10(291)10(311
4
)10(1141
)10(1216434434434
43
veff
3
3
4
3
4
3
2
2
4
2
4
2
1
1
4
1
4
1
4
)()()(
)(
v
xuc
v
xuc
v
xuc
yuc
effv
Our effective degrees of freedom considering all our uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 53
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53
cm
kuU
Calculating the expanded uncertainty
Level of
Confidence
(percent)
Coverage
factor
k
6827 1
90 1645
95 1960
9545 20
99 2576
9973 3
k is the coverage factor
bull How confident should you be with your measurement results
(68 95 99)
bull 95 confidence is commonly accepted as appropriate
bull Um expresses the uncertainty expanded from a single standard
deviation of 68 to uncertainty value with a higher confidence
bull For a large population with a normal distribution 95 coverage
is calculated by k with a value of 196
(or sometimes 2 for convenience ndash giving 9545)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 54
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54
Adjusting k for a smaller set of measurements or samples
bull Adjusting k is done using the
studentsrsquo t distribution table
bull A coverage factor adjustment
is needed because our data
set had a fewer number of
values rather than a larger set
(such as 20 50 or 100)
bull The table lists the proper
coverage factor for populations
with smaller degrees of
freedom
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
For our example with the effective degrees of freedom (Veff) of 52
a coverage factor of 257 expands uc to a value with 95 confidence
(compared to 196 for an infinite set of measurementssamples)
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 55
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55
cm
kuU
Expanded measurement uncertainty calculation
572Um1216 mA
U m 3126 mA
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 56
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56
Our overall uncertainty budget
Source of
Uncertainty Type Ui
Uncertainty
Value
(Amps)
Sensitivity
Coefficient
Probability
Distribution
Coverage
Factor
Standard
Uncertainty
(Amps)
Degrees
of
Freedom
Repeatability A u1 11410-3 1 Normal 1 11410-3 4
Calibrator B u2 710-3 1 Normal 258 2710-3
Resolution B u3 510-3
1 Rectangular 2910-3
Current
Measurement Combined uC - -
Assumed
Normal - 12110-3 52
Current
Measurement Expanded Um
312610-3 - Assumed
Normal 257 - 52
3
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 57
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57
mavg UII
Final results -
bull The final measurement value including the
measurement uncertainty from the series of DMM
measurements of the calibrator
AmpsI 0410 0031 At a level of confidence of 95
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 58
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58
What if more measurements were taken does that improve the uncertainty
Increased degrees of freedom
Veff = 5 10 20 or 100
Causes marginal improvements
in k and in
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull 9 measurements Veff = 103
minus k = 223 = 27 mA (4 mA better)
bull 17 measurements Veff = 207
minus k = 209 = 25 mA (2 mA better)
bull 78 measurements Veff = 1009
minus k = 1984 = 24 mA (1 mA better)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
U m
U m
U m
So improves only 7 mA by taking
73 more measurements U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 59
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59
Does improving beyond plusmn31 mA by taking more measurements have any practical value
Whatrsquos the value of increasing
Veff from 5 to
The test tolerance is plusmn250 mA
bull 5 measurements Veff = 52
minus k = 257 = 31 mA
bull With a = 31mA
the test ratio is already 81
(TUR = Test Spec divide Total Uncertainty
025A divide 31mA = 806)
Fraction p in percentDegrees of
freedom 6827 90 95 9545 99 9973
1 184 631 1271 1397 6366 2358
2 132 292 43 453 992 1921
3 12 235 318 331 584 922
4 114 213 278 287 46 662
5 111 202 257 265 403 551
6 109 194 245 252 371 49
7 108 189 236 243 35 453
8 107 186 231 237 336 428
9 106 183 226 232 325 409
10 105 181 223 228 317 396
20 103 172 209 213 285 342
50 101 168 201 205 268 316
100 1005 166 1984 2025 2626 3077
1 1645 196 2 2576 3
U m
U m
AmpsI 0410 0031
So to satisfy a minimum test ratio of 41
5 measurements are more than adequate
U m
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 60
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60
Questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 61
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61
Making The Calculation Of Measurement Uncertainty Simpler
What can you do to automate this
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 62
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62
Automation alternatives
bull A custom program
designed for a specific
requirement
bull A custom spreadsheet for
analysis
bull A commercial metrology
based software package
such as
Flukersquos METCAL Plus
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 63
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63
METCAL automates the uncertainty calculations
Post test summary of
10000A 50Hz
Including
5 reading average
Calculated combined
standard uncertainty
How does this work
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 64
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64
METCAL manages amp analyses the uncertainties
Number of Measurements = 5
Value 1 = 1007
Value 2 = 1001
Value 3 = 1002
Value 4 = 1004
Value 5 = 1006
UUT Indicated = 1004
Standard Deviation = 002549509757
Standard uncertainty = 001140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4
System Actual = 10
System Accuracy = 0008
Confidence interval of spec = 258
1 Sigma Spec = 0003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
UUT Resolution = 001
Resol Standard Uncertainty = 0002886751346
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200
Combined Std Uncertainty = 001216490061
Effective Deg of Freedom = 5186506
Standard Uncertainty = 001207040471
Coverage Factor = 2567104753
Expanded Uncertainty = 0031263794
Calculated
Total
Uncertainty
Repeatability
Uncertainty
Calibrator
Uncertainty
Resolution
Uncertainty
Measurement
Details
With METCAL the user configures
bull Specific statistics used
bull Confidence Coverage
bull Number of measurements
bull Accuracy of the standard
In the cal or test procedure you also specify test parameters
bull Test point
bull UUT resolution
In the test process METCAL provides the uncertainty details (our example is shown to the right)
Details are permanently stored in the data base They accessible for reports amp future analysis
METCAL Data for
our example
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 65
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65
ldquoAutomationrdquo ndash some words of wisdom
bull Remember it is always the metrologistrsquos responsibility to
insure proper calculation of measurement uncertainty
minus Every lab has unique characteristics which must be supported
minus Configuring the measurement characteristics is also unique
minus Defining the specific error budget for the test
minus Configuring the specific measurement uncertainty parameters
bull There should be definite information to support answering
any auditorrsquos questions
bull Keep records of the procedurersquos measurement design with
an uncertainty error budget
bull Be able to demonstrate the reasonableness of the testrsquos
uncertainties
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 66
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66
Benefits of METCAL automation
bull Automation simplifies a structured calculation process
bull Usable for manual semi automated or fully automated testing methods
bull METCAL provides flexibility to customize the calculation process amp factors
bull METCALrsquos database stores all the information for future reference
bull Report writing flexibility permits properly configured certificates and data summaries
bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 67
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67
Automation questions
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 68
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68
Conclusion amp Review ndash What have we done
bull Topics
minus Measurement uncertainty amp why it is important
minus How measurement uncertainty obtained
minus Examples on measurement uncertainty amp calibrating DMMs
minus Benefits of automating
bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements
bull Measurement results are considered incomplete without a quoted uncertainty
bull Calculations usually require a statistical process on multiple measurements for each test
bull Automation can be a valuable support for measurement uncertainty calculations
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 69
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69
Obtain a copy of the GUMs amp
other references for details
ANSINCSL Z5402-1997 (R2002) US
Guide to Expression of Uncertainty in
Measurement httpwwwncsliorg and find it in the store
under NCSLI publications
NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines
contentshtml
Where to go from here
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 70
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70
For more information (1) -
bull Chapters 20-22 on Statistics amp
Uncertainty in the text book
Calibration Philosophy in
Practice 2nd Edition
bull Flukersquos Training Course ndash Cal Lab
Management for the 21st Century
bull Various reference material under
technical papers at the resource
library on Flukersquos web site
httpwwwflukecom
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 71
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71
For more information (2) -
bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg
bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom
bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl
bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 72
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72
Still more references (3)
bull NCSL International RP-12 - Determining amp
Reporting Measurement Uncertainties httpswwwncsliorg
bull NIST Website Essentials of expressing
measurement uncertainty
httpphysicsnistgovcuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 73
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73
Questions
22
3
2
2
2
1 nc uuuuu
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 74
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74
Fluke Calibration Web Seminar Series
For information amp reservations to attend our
seminars go to wwwflukecalcom click
on the menu selection ldquoEvents amp
Trainingrdquo and click on the ldquoWeb
Seminarsrdquo selection and again click on
the desired seminar selection
Our Seminar Topics Include
bull Precision Measurement Techniques
bull Oscilloscope Calibration
bull General Metrology
bull Temperature Calibration
bull Metrology Software
bull RF Calibration
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 75
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75
Calibration and metrology training
bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)
minus MET-301 Advanced Hands-on Metrology (new in 2007)
minus MET-302 Hands-on Metrology Statistics (new in 2009)
minus Cal Lab Management for the 21st Century
minus Metrology for Cal Lab Personnel (A CCT prep course)
minus METCAL Database and Reports
minus METCAL Procedure Writing
minus METCAL Advanced Programming Techniques
minus On-Site Training
minus Product Specific Training
bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training
minus METCAL Procedure Development Web-Based Training
bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration
minus Precision Electrical Measurement
minus Measurement Uncertainty
minus ACDC Calibration and Metrology
minus Metrology for Cal Lab Personnel (A CCT prep course)
bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training
minus METCAL-CBTPW Computer-Based Training (new in 2007)
minus Cal-Book Philosophy in Practice textbook More information
wwwflukecalcomtraining
Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20
discount off any Fluke calibration training course
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
For material related to this session visit our web site
httpwwwflukecom
For any questions or copies of this presentation
email inquiries to calibrationseminarsflukecom
Page 76
copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76
THANK YOU
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