© SPEX CertiPrep 2011 Vanaja Sivakumar, PhD Vice President, Manufacturing 10/27/2011 Identifying and Quantifying the Uncertainty Associated with Instrumental Analysis
© SPEX CertiPrep 2011
Vanaja Sivakumar, PhDVice President, Manufacturing
10/27/2011
Identifying and Quantifying the Uncertainty Associated with Instrumental Analysis
© SPEX CertiPrep 2011
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Topics Covered
Precision & AccuracyPrecision & Accuracy
Confidence LimitsConfidence Limits
Errors in MeasurementsErrors in Measurements
Uncertainty in MeasurementsUncertainty in Measurements
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True Value (TV)
MethodMethod
InstrumentInstrument
Skill of the OperatorSkill of the Operator
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Sources of Error (E)
Determinate (Systematic)
• Improper Calibration• Personal Errors• Error in Methods
Indeterminate (Random)
• Experimental Errors
Spurious
• Human Errors• Instrument Failures
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Central tendency
Not an absolute value
Estimate of true value
Average (0)
Sum of the results of each measurement
number of measurementsAverage (0) = 0 =
…
True value is described by average and standard deviation
or
Average is:
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Standard Deviation (s)
0 ± 1.96 s at 95% Confidence Interval
0 ± 2.94 s at 99% Confidence Interval
Standard Deviation (s)
You can narrow your estimate of True Value using standard deviation
Can also be described as sigma,
True Value can be expressed as:
0
1
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Confidence Intervals
The Average gives an estimate of the true value
The Confidence Interval can pinpoint the range within which the true value lies
– Also called Interval Estimate or Confidence Limit
Confidence Intervals are usually expressed at 90%, 95% or 99%
-3 -2 - 0 2 3
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Accuracy & Precision
PrecisionPrecision• Reproducibility of the method • Evaluated by Average Deviation,
Variance and Standard Deviation
AccuracyAccuracy• Correctness of the measurement
in relation to the true value• (TV=0 ± E) or Measured value ± E
True Value ( ) = 8.50
7
8
9
10
1 2 3 4 5 6 7 8
Valu
es
Measurement #
Good Accuracy (0 = 8.55)
7
8
9
10
1 2 3 4 5 6 7 8
Valu
es
Measurement #
Poor Accuracy (0 = 7.39)
7
8
9
10
1 2 3 4 5 6 7 8
Valu
es
Measurement #
Good Precision (0 = 8.55)
7
8
9
10
1 2 3 4 5 6 7 8
Valu
es
Measurement #
Bad Precision (0 = 8.55)
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Evaluation of Precision
Key Word Symbol Equation Description
Average Deviation
AD
The smaller the AD, the more precise the measurements
Not an accurate measure of precision
Variance V
Indicates the spread of measurements around the data
Better measure of precision
StandardDeviation s The amount of variation or
spread from the True Value
AD
s 0
1
0
0 2
1
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Glossary Review
Key Word Symbol Equation
Average 0
Standard Deviation s
0 …
s 0
1
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Uncertainty
Uncertainty is a parameter associated with the result of measurement, that characterizes the dispersion of the values that could be reasonably attributed to measured value
Error vs. Uncertainty:
– Error: Usually can’t be estimated
– Uncertainty: Can be estimated
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Uncertainty Estimation
Determine what is to be measuredDetermine what is to be measured
Outline the various processesOutline the various processes
Identify sources of uncertaintyIdentify sources of uncertainty
Estimate uncertainties from each sourceEstimate uncertainties from each source
Combine and expand all uncertaintyCombine and expand all uncertainty
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Types of Uncertainty
Type AType A• Usually associated with
repeated measurements• Type A uncertainty is
expressed as:
Type BType B• Based on scientific judgment
made from previous experience, manufactures’ specifications, etc.
• Three common models are:
Where s is standard deviation and n is the number of replicates
o Rectangularo Triangularo Normal
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Type B: Rectangular Distribution
Use this when a certificate or other specification gives limits without specifying a level of confidence
Normalizing factor for converting to Standard Uncertainty is
X
3
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Type B: Triangular Distribution
This is used when distribution is symmetric and when values close to the target value are more likely than near the boundaries
Example: Uncertainty associated with volumetric glassware
Normalizing factor for converting to Standard Uncertainty is
6
X
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Type B: Normal Distribution
This is used when an estimate is made from repeated observations of a randomly varying process and an uncertainty is associated with a certain confidence interval.
Example: A calibration certificate with stated level of confidence.
Normalizing factor for converting to Standard Uncertainty is
X
2
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Combined Uncertainty (uc)
We will be using the following model to combine overall uncertainties from all of the processes:
We will use the following model for calculating interim uc within a process:
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Expanded Uncertainty (U)
The Expanded Uncertainty defines the interval within which lies the value of the measurand
U ku
To calculate, multiply the combined standard uncertainty with a Coverage Factor (k):
Coverage factor k depends on the level of confidence and the degree of freedom:
k= 2 for 95% confidence level
TrueValue X 0 U
m1
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Glossary Review
Key Word Symbol Equation
Average 0
Standard Deviation s
Combined Uncertainty
Expanded Uncertainty
0 …
s 0
1
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Review of the Steps
Determine what is to be measuredDetermine what is to be measured
Outline the various processesOutline the various processes
Identify sources of uncertaintyIdentify sources of uncertainty
Estimate uncertainties for each sourceEstimate uncertainties for each source
Combine all the components and expandCombine all the components and expand
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Process Outline for Analysis of Au
Components: Symbol Value
NIST SRM conc• Weighing• Flask• SRM value
Stdp 100.225 mg/L
Sample Dilution• Pipette• Flask
Crmf 100 fold
SRM Measured conc Stdm 100.985 mg/L
CRM Measured conc Crmm 100.797 mg/L
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Determination of concentration
C Crm Std Crm
Std
Where:
Cau = Concentration of Au in sample
Crmm = ICP measured concentration of sample
Crmf = Dilution factor for sample
Stdp = Prepared concentration of SRM
Stdm = ICP measured concentration of standard
C100.797 100.225 100
100.985
10,004mg/L
Component Symbol ValueNIST SRM conc
• Weighing• Flask• SRM value
Stdp 100.225mg/L
Sample Dilution• Pipette• Flask
Crmf 100 fold
SRM Measured conc StdM 100.985 mg/L
CRM Measured conc CrmM 100.797 mg/L
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ICP-Sample measurement (Crmm)
ICP-SRM measurement (Srmm)
Tolerance Temp
Vol
COA
Prep. SRM (Stdp)
Mass
Uncertainty Sources
The aim is to identify all major uncertainty sources and to understand their effect on the analyte and its uncertainty
Uc(Cau)
Pipette
Flask
Sample dil (Crmf)Tolerance Temp
Tolerance Temp
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Uncertainty from SRM - U(Stdp)
Weighing on the balanceWeighing on the balance
500 mL Flask
SRM Certification
Used NIST SRM 3121 for calibration
To prepare the SRM:
– Weighed 5.067 g
– Diluted to 500 mL in a volumetric flask
There are 3 uncertainties associated with this process:
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U(Stdp) – From the Balance
Weighing on the balanceWeighing on the balance 500 mL Flask SRM
Certification
1√3
Device Value (V) Standard u Combined u(uc)
Relative u( )
Balance 5.0670 g 8.165 x 10-5 8.165 x 10-5 1.611 x 10-5 2.597 x 10-10
UncertaintyfrombalancecalibrationListed
NormalizationFactorforRectangularDistribution
0.00013
5.774 10
StandardUncertainty TareWeight GrossWeight
5.774 10 5.774 10
8.165 10
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U(Stdp) – From the Flask
Weighing on the balanceWeighing on the balance
SRM Certification
500 mL Flask
There are 2 uncertainties associated with the dilution:
uListedTolerance
60.26
.
.
uThermalCoefficientofWater TempVariation Volume
NormalizationFactorforRectangularDistribution
2.1x10 oC/mL 3oC 500mL 3
, triangular
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U(Stdp) – From the Flask (cont.)
Device Value(V) uvolume utemp
Combined u(uc)
Relative u( )
Flask 500 mL 0.08165 0.1819 0.19935 0.0003987 1.5897 x 10-7
Weighing on the balanceWeighing on the balance
SRM Certification
500 mL Flask
CombinedUncertaintyu
0.08165 0.1819
0.19935 Relative
0.19935500
0.0003987
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U(Stdp) – From the SRM Cert.
Weighing on the balanceWeighing on the balance 500 mL Flask
DeviceCertified
SRM Value(V)
Expanded u Coverage Factor Standard u Relative u
COA 9.89 mg/g 0.02 2 0.01 1.0111 x 10-3 1.0224 x 10-6
SRM Certification
Standardu Expanded
CoverageFactor
0.022
0.01
Relativeu Standard
V
0.019.89
1.011x10
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Total Uncertainty Due to Stdp
Add each uncertainty component from the preparation of the standard…
Balance 1.61 x 10-5 2.597 x 10-10
500 mL Flask 0.0003987 1.5897 x 10-7
SRM Certification 0.001011 1.0224 x 10-6
Total: 1.1816 x 10-6
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Uncertainty from Sample Dilution -U(Crmf)
500 mL FlaskPipette
To prepare the sample:
– A 5 mL of the sample was diluted to 500 mL
There are 2 uncertainties associated with Crmf:
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U(Crmf) – From the Pipette
uListedu
NormalizationFactorforTriangularDistribution
0.016
.
.
Pipette 500 mL Flask
uThermalCoefficientofWater TempVariation Volume
NormalizationFactorforRectangularDistribution
2.1x10 oC/mL 3oC 5mL 3
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Pipette 500 mL Flask
CombinedUncertainty
0.004082 0.1819x10
0.004469
RelativeuCombined
V
0.0044695
.
U(Crmf) – From the Pipette (cont.)
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Pipette
U(Crmf) – From the Flask
Flask
uListedu
NormalizationFactor
0.26
0.08165
.
ThermalCoefficientofWater TempVariation Volume NormalizationFactorforRectangularDistribution
2.1x10 oC/mL 3oC 500mL 3
TypeB, triangular
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U(Crmf) – From the Flask (cont.)
Pipette Flask
Value(V) utol utemp
Combined u(uc)
Relative u( )
Pipette 5 mL 0.004082 1.819 x 10-3 0.004469 0.0008938 7.989 x 10-7
Flask 500 mL 0.08165 0.1819 0.19935 0.0003987 1.5896 x 10-7
Total 9.5793 x 10-7
CombinedUncertainty
0.08165 0.1819
0.19935
RelativeuuV
0.19935500
0.0003987
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ICP Measurement for standard-U(Stdm)
Measurement Stdm conc (mg/L)
1 101.260
2 100.853
3 100.985
4 101.542
5 100.914
6 100.685
7 101.251
8 100.720
9 100.654
0 100.985
s 0.29397
0
1s 0.29397mg/L
0… 100.985mg/L
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Standard Concentration Measurement-U(Stdm)
0.293979
0.09799
0.09799100.985
0.00097035
Value(V) sd # reps uc
Stdm 100.985 0.29397 9 0.09799 0.00097035 9.4158 x 10-7
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ICP Measurement for sample-U(Crmm)
Measurement Crmm conc (mg/L)
1 100.200
2 100.781
3 101.054
4 101.108
5 100.775
6 100.854
7 100.578
8 101.075
9 100.752
0 100.797
s 0.3065
0.3065mg/Lx 0
n 1s
0…
100.797mg/L
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Sample Concentration Measurement-U(Crmm)
Value(V) sd # reps Uc
Crmm 100.797 0.3065 9 0.102165 0.0010136 1.0273 x 10-6
usd9
0.30659
0.102165
RelativeuuV
0.102165 100.797
0.0010136
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Concentration And Uncertainty of Au Solution By ICP-OES
Symbol Input
Stdp Uncertainty SRM prep 1.1816 x 10-6
Crmf Uncertainty sample prep 9.5793 x 10-7
StdmUncertainty SRM
measurement 9.4158 x 10-7
CrmmUncertainty sample
measurement 1.0273 x 10-6
Total uc 4.10843 x 10-6
0.002026927
Uc (Cau) 0.002027 x (10004) 20.2771
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The Certified Value And The Expanded Uncertainty U for Gold Solution By ICP Analysis
The ICP value for Gold = 10,004 ± 41 mg/L
2 20.27740.554mg/L
CoverageFactorfor95%CI TotalCombinedUncertainty uc
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References
Kocherlakota, N, Obenauf, R, “A statistical approach to reporting uncertainty”
Quantifying Uncertainty Measurements, Eurachem/CETAC Guide
Kocherlakota, N, Obenauf, R, “Quantitative Calculation of Uncertainty Associated with Gold Reference Material,” To be published by NIST
British Standards 1797:1952
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