Analisa Ketidakpastian - Statistik

PART 1

Introduction

and Underlying Concepts

Uncertainty Analysis – What it is

There is no such thing as a perfect

measurements. All measurements of a variable

contain inaccuracies.

The analysis of the uncertainties in

experimental measurements and results is a

powerful tool, particularly when it is used in

the planning and design of experiments

Although it may be possible to decrease an

uncertainty by improved experimental method

or the careful use of statistical technique to

reduce the uncertainty, it can never be

eliminated

Issues of Analysis

Systematic and Random Uncertainties

Issues of Analysis

Systematic Uncertainties

Offset uncertainty

Clearly there is a problem here:

the boiling point of water should be very close to 100.0 oC while the melting point should be very close to 0.0 oC

There is an offset uncertainty with the temperature

measuring system of about 7.5 oC

Possible causes are inherent to measurement device

(such as low battery, malfunctioning digital meter,

incorrect type of thermocouple, etc)

Issues of Analysis

Systematic Uncertainties

Gain uncertainty

(mb - mc) versus mc

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 20.00 40.00 60.00 80.00 100.00 120.00

mc (g)

mb

- mc

(g)

Issues of Analysis

Random Uncertainties

Random uncertainties produce scatter in observed

values.

The cause :

o limitation in the scale of the instrument

resolution uncertainty due to rounding up of

measured value

o reading uncertainty

o random uncertainty due to environmental factor

(electrical interference, vibration, power supply

fluctuation, Brownian motion of air molecule,

background radiation, noise, etc)

Use statistical technique to get an estimate of the

probable uncertainty and to allow us to calculate the

effect of combining uncertainties

Issues of Analysis

True Value, Accuracy and Precision

Statistical Basics on Uncertainty

Data


Sample and Population


Frequency Distribution


Histogram and Probability Distribution


Mean and Standard Deviation

Sample Population

Mean

Standard

Deviation

Variance = (Standard Deviation)2

Variance


Standard Error (of the Mean)

If standard deviation of population (x )of is known :

In a sampling with n repeated measurement (sampling

size = n), the standard error is determined as:

Standard

Error =

If standard deviation of population of is unknown, then

it is estimated from standard deviation of sample (sx)

for infinite population

For finite population

with size of N

Standard

Error = or


Gaussian (Normal) Distribution Function


Probability of Gaussian Distribution


Standard Gaussian Distribution

PART 2

Determining Measurement

Uncertainty

Data Handling – Rejection of Outliers

Rejection of outliers is more acceptable in areas of

practice where the underlying model of the process

being measured and the usual distribution of

measurement error are confidently known

You want some way to identify what observations in

your data set need closer study. It's not appropriate to

simply throw away or delete an observation; you must

keep it around to look at later. The picture is as follows.

Filtering Process (Selection and Rejection of Data)


Filtering Process (Selection and Rejection of Data) A sensitive subject and one that can bring out

strong feeling amongst experimenters:

o One argue : All data are equal no

circumstances in which the rejection of

data can be justified

o Another argue : there as those that ‘know’

that a set of data is spurious and reject it

without a second thought

Expert judgment confidence level

Statistical test :

o Chauvenet’s criterion P = 1 – 1/(2n)

o Peirce’s criterion

o - σ criterion = 2, 3, …

chauvenetscriterion.pdf


Chauvenet’s criterion

Wide acceptance method which defines an acceptable

scatter around the mean value from a given sample of

n readings from the same parent population

All points should be retained that fall within a band

around the mean value that corresponds to a

probability of 1 – 1/(2n)


Chauvenet’s criterion

Dealing with Uncertainty

Quoting the Uncertainty

After making repeated measurement of a

quantity, there are four important steps to take in

quoting the value of the quantity:

1. Calculate the mean of the measured values

2. Calculate the uncertainty in the quantity,

making clear the method used. Round the

uncertainty to one significant figure (or two if

the first figure is a ‘1’)

3. Quote the mean and uncertainty to the

appropriate number of figures

4. State the units of the quantity

Dealing with Uncertainty

Uncertainty statement

Absolute uncertainty

o With unit of the quantity

) of(unit XUX x

Fractional uncertainty

o no unit

X

U Xy uncertaint fractional

Percentage uncertainty

o no unit

%100y uncertaint percentage X

U X

Determining Uncertainty – Single

Measurement

If only one measurement is made, the uncertainty

is generally determined from the instrument

resolution of reading scale (resolution

uncertainty)

Uncertainty = half of the smallest resolution of reading

Example :

A length of a stick is measured with a rule with resolution

of reading scale 1 mm. The reading is 361 mm, then the

length should be quoted as:

L = (361.0 0.5) (mm)

Determining Uncertainty – Repeated

Measurement of Single Quantity

Simple Method - 1

n

XX minmax

tmeasuremen ofnumber

range mean in y uncertaint

Example:

)/( 8.341mean smc

(m/s) 75.3385.345range cR

(m/s) 875.08/7yuncertaint cU

(m/s) 100.0093.418

(m/s) 9.08.341

2

c



Simple Method - 2

Example:

)/( 8.341mean smc

4.11

8.3417.342...8.3414.3428.3415.341deviation Total

cci

425.18/4.11deviationMean y Uncertaint

(m/s) 100.0143.418

(m/s) 4.18.341

2

c



Statistical Approach to variability in data

mean theoferror tandard mean in y uncertaint s

n

sxx xxxx kUU limits confidence



(s) 006.050

043.0 mean, theoferror Standard

(s) 043.0 deviation, Standard

(s) 604.0 Mean,

t

ts

t

With 95 % level of confidence : (s) 012.02 ttU

Then : (s) 012.0604.0 t

Combining Uncertainty – Uncertainty

Propagation

An experiment may require the determination of several

quantities which are later to be inserted into an equation.

The uncertainties in the measured quantities combine to

give an uncertainty of the calculated value

The combination of these uncertainties is sometimes called

the propagation of uncertainty or error propagation

V

m

measured quantity

with uncertainty

measured quantity

with uncertainty

calculated quantity

with propagation of

uncertainty

Combining Uncertainty – Simple Method

Most straightforward method and requires only simple

arithmetic.

Each quantity in the formula is modified by an amount equal

to the uncertainty in the quantity to produce the largest

value and the smallest value

Example :

In an electrical experiment, the current through a resistor was

found to be (2.5 ± 0.1) mA and the voltage across the resistor

(5.5 ± 0.3) V. Determine the resistance of the resistor R and the

uncertainty UR !

1020.2A 102.5

V 5.5 3

3-I

VR

1042.2A 102.4

V 8.5 3

3-maxR

1000.2A 102.6

V 2.5 3

3-minR

10210.02

3minmax RR

102.02.2 3R

Combining Uncertainty – Partial

Differentiation

Based on differentiation of

function of several variables

bb

Va

a

VV

baVV

),(

baV Ub

VU

a

VU

bb

Va

a

VV

Uncertainty propagation:

Properties:

baVbaV

baVbaV

b

b

a

a

V

VabV

b

b

a

a

V

VbaV

/

Sum:

Difference:

Product:

Quotient:

Combining Uncertainty – Partial

Differentiation

Example :

The temperature of (3.0 ±0.2) x 10-1 kg of water is raised by (5.5

± 0.5) oC by heating element placed in the water. Calculate the

amount of heat transferred to the water to cause this

temperature rise !

J 1088.4

)(0.5)(0.3)(4186 )(0.02)(4186)(5.5

and

J 6907)5.5)( )(4186 (0.3

mcUUcU

mcQ

cm

Q

UQ

Um

QU

mcQ

mQ

mQ

The value of c = 4186 J kg-1 oC-1

is assumed to be constant

(neglecting its uncertainty)

J 101.19.6 3Q

Combining Uncertainty – Statistical

Approach

2

2

2

2

2

2

2

2

2

),(

baV

baV

b

V

a

V

b

V

a

V

baVV

Taking uncertainty of the mean to relate to standard error of

the mean and partial differential principle

Uncertainty in Linear Fitting of X-Y Data

Least Square Method in Linear Fitting

22ii

iiii

xxn

yxyxnm

22

2

ii

iiiii

xxn

yxxyxc


In reality for each value of x, the corresponding y

has some uncertainty

contribute to the uncertainty of m and c


Uncertainty of m

22

ii

m

xxn

n

Uncertainty of c

22

2

ii

i

m

xxn

x

where

2

2

1

cmxy

nii

PART 3

More on Measurement

Uncertainty

General Uncertainty Analysis

Consider a general case in which an experimental result, r, is

a function of J measured variable Xi

JXXXrr ,...,, 21

Then, the uncertainty in the result is given by :

2

1

2

2

2

2

2

2

2

2

1

2 ...21 i

J

i i

X

J

XXr UX

rU

X

rU

X

rU

X

rU

J

iX

i

i

XU

X

r

i variablemeasured in they uncertaint

tcoefficieny sensitivit absolute

Note : all absolute uncertainties (UX) should be expressed with

the same level of confidence


Nondimensionalized forms:

222

2

2

2

2

2

1

2

1

1

2

2

...21

j

X

j

jXXr

X

U

X

r

r

X

X

U

X

r

r

X

X

U

X

r

r

X

r

U j

Note : factorion magnificaty uncertaint

i

i

i UMFX

r

r

X

oncontributi percentagey uncertaint12

22

UPC

r

U

X

U

X

r

r

X

r

i

X

j

i i


Example:

A pressurized air tank is nominally at ambient temperature (25 oC).

Using ideal gas law, how accurately can the density be determined

if the temperature is measured with an uncertainty of 2 oC and the

tank pressure is measured with a relative uncertainty of 1%?


Uncertainty analysis:

222

2222

2

2

2222222

zero is that assuming

1

1

1

,,

T

U

p

UU

UT

U

R

U

p

UU

RT

p

RT

pT

T

T

RT

p

TR

pR

R

R

RT

p

p

p

T

U

T

T

R

U

R

R

p

U

p

pU

TRpRTp

Tp

RTRp

TRp


Uncertainty analysis:

%2.1012.0)298/2(01.0

01.0

298

2

29827325

2 2

22

2

UU

p

U

T

U

KT

KCU

p

T

o

T

Detailed Uncertainty Analysis

X1

B1,P1 X2

B2,P2

Xj

Bj,Pj

1 2 j ….

….

r=r(X1,X2, …, Xj)

r

Br,Pr

Elemental error sources

Individual measurement system

Measurement of individual

variables

Equation of result

Experimental result

B = bias (systematic uncertainty)

P = precision (random) uncertainty

Detailed Uncertainty Analysis

The uncertainty in the result is:

222

rrr PBU

Systematic (bias) uncertainty:

1

1 11

222 2J

i

J

ik

ikki

J

i

iir BBB

J

i

iir PP1

222

Precision (random) uncertainty:

Correlated systematic

uncertainty

Systematic Uncertainty

Systematic error can be determined and eliminated by

calibration only to a certain degree (A certain bias will

remain in the output of the instrument that is calibrated)

In the design phase of an experiment, estimate of systematic

uncertainty may be based on manufacturer’s specifications,

analytical estimates and previous experience

As the experiment progress, the estimate can be updated by

considering the sources of elemental error:

o Calibration error: some bias always remains as a result

of calibration since no standard is perfect and no

calibration process is perfect

o Data acquisition error: there are potential biases due to

environmental and installation effects on the transducer

as well as the biases in the system that acquires,

conditions and stores the output of the transducer

o Data reduction errors: biases arise due to replacing data

with a curve fit, computational resolution and so on

Random Uncertainty Analysis

Random uncertainty can be determined with various ways

depending on particular experiment:

o Previous experience of others using the same/similar

type of apparatus/instrument

o Previous measurement results using the same

apparatus/instrument

o Make repeated measurement

When making repeated measurement, care should be taken

to the time frame that required to make the measurement:

o Data sets should be acquired over a time period that is

large relative to the time scale of the factors that have a

significant influence on the data and that contribute to

the random errors

o Be careful of using a data acquisition system

Random Uncertainty Analysis

∆t

Time, t

Y

Failure to determine random

uncertainty due to inappropriate

data acquisition

Some Detail Approach/Guidelines

Abernethy approach (1970-1980):

o Adapted in SAE, ISA, JANNAF, NRC, USAF, NATO

estimate confidence 99%for

estimate confidence 95%for 2/122

rrADD

rrRSS

tSBU

tSBU

Coleman and Steele approach (1989 renewed 1998):

o Adapted in AIAA, AGARD, ANSI/ASME 222

rrr PBU

1

1 11

222 2J

i

J

ik

kiikki

J

i

iir BBBB

1

1 11

222 2J

i

J

ik

kiSikki

J

i

iir PPPP

Some Detail Approach/Guidelines

ISO Guide approach (1993):

o Adapted by BIPM, IEC, IFCC, IUPAC, IUPAP, IOLM

o Using a “standard uncertainty”

o Instead of categorizing uncertainty as systematic and

random, the “standard uncertainty”values are divided

into type A standard uncertainty and type B standard

uncertainty

o Type A uncertainties are those evaluated “by the

statistical analysis of series of observations”

o Type B uncertainties are those evaluated “by means

other than the statistical analysis of series of

observations”

NIST Approach (1994):

o Use “expanded uncertainty” U to report of all NIST

measurement other than those for which Uc has

traditionally been employed

o The value of k = 2 should be used. The values of k other

than 2 are only to be used for specific application

References

[1] Montgomery, D.C., & Runger, G.C, Applied Statistics and Probability for Engineers 3rd

Ed., John Wiley and Sons, Inc., New York, (2003)

[2] Harinaldi, Prinsip-prinsip Statistik Untuk Teknik dan Sains, Erlangga, (2005)

[3] Kirkup, L, Experimental Method: An Introduction to the Analysis and Presentation of

Data, John Wiley and Sons, Australia, Ltd., Queensland, (1994)

[4] Coleman, H.W and Steele, W.G., Experimentation and Uncertainty Analysis for Engineer

2nd Ed., John Wiley and Sons, Inc., New York, (1999)

[5] Doebelin, E.O, Engineering Experimentation, Planning Execution, Reporting, McGraw-

Hill Book Co., New York, (1995)

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