LM01-Aspek Statistik Pengukuran-Analisis Ketidakpastian

8/11/2019 LM01-Aspek Statistik Pengukuran-Analisis Ketidakpastian

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UNCERTAINTY ANALYSIS IN

MEASUREMENT(Review of Stat ist ical Aspects in Engineering Measurement)

Prof. Dr. Ir. Harinaldi, M.EngDepartment of Mechanical Engineering

Faculty of Engineering, University of Indonesia

App l ied Flow Measurement and Visual izat ion Lecture Module


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PART 1

Introduction

and Under lying Concepts


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Uncertainty Analysis What it is

There is no such thing as a perfect

measurements. All measurements of a variablecontain inaccuracies.

The analysis of the uncertainties in

experimental measurements and results is a

powerful tool, particularly when it is used inthe 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


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Issues of Analysis

Systematic and Random Uncertainties


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Issues of Analysis

Systematic Uncertainties

Offset uncertainty

Clearly there is a problem here:

the boiling point of water should be very close to 100.0o

C while the melting point should be very close to 0.0o

C 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)


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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-m

c

(g)


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Issues of Analysis

Random Uncertainties

Random uncertainties produce scatter in observedvalues.

The cause :

o limitation in the scale of the instrument

resolution uncertainty due to rounding up of

measured valueo 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


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Issues of Analysis

True Value, Accuracy and Precision


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Statistical Basics on Uncertainty

Data


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Sample and Population


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Frequency Distribution


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Histogram and Probability Distribution


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Mean and Standard Deviation

Sample Population

Mean

Standard

Deviation

Variance = (Standard Deviation)2

Variance


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Standard Error (of the Mean)

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

In a sampling with nrepeated measurement (sampling

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

StandardError =

If standard deviation of population of is unknown, thenit is estimated from standard deviation of sample (sx)

for infinite population

For finite population

with size of N

Standard

Error = or


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Gaussian (Normal) Distribution Function


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Probability of Gaussian Distribution


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Standard Gaussian Distribution


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PART 2

Determining Measurement

Uncertainty


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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)


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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 levelStatistical test :

o Chauvenets criterion P = 11/(2n)

o Peirces criterion

o - riterion = 2, 3,
http://localhost/var/www/apps/conversion/tmp/scratch_3/chauvenetscriterion.pdfhttp://localhost/var/www/apps/conversion/tmp/scratch_3/chauvenetscriterion.pdf


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Chauvenets 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 11/(2n)


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Chauvenets criterion


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Dealing with Uncertainty

Quoting the Uncertainty

After making repeated measurement of aquantity, there are four important steps to take in

quoting the value of the quantity:

1. Calculate the mean of the measured values2. 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


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Dealing with Uncertainty

Uncertainty statement

Absolute uncertainty

o With unit of the quantity

)of(unit XUX x

Fractional uncertaintyo no unit

X

UXyuncertaintfractional

Percentage uncertainty

o no unit

%100yuncertaintpercentage

X

UX


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

length should be quoted as:

L = (361.0 0.5) (mm)


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Determining Uncertainty Repeated

Measurement of Single Quantity

Simple Method - 1

n

XX minmax

tmeasuremenofnumber

rangemeaninyuncertaint

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


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Simple Method - 2

Example:

)/(8.341mean smc

4.11

8.3417.342...8.3414.3428.3415.341deviationTotal

cci

425.18/4.11deviationMeanyUncertaint

(m/s)100.0143.418

(m/s)4.18.341

2

c


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Statistical Approach to variability in data

meantheoferrortandardmeaninyuncertaint s

n

sxx xxxx kUU limitsconfidence


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(s)006.050

043.0mean,theoferrorStandard

(s)043.0deviation,Standard

(s)604.0Mean,

t

ts

t

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

Then : (s)012.0604.0 t


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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 ofuncertainty


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Combining Uncertainty Simple Method

Most straightforward method and requires only simple

arithmetic.

Each quantity in the formula is modified by an amount equalto the uncertainty in the quantity to produce the largest

value and the smallest value

Example :

In an electrical experiment, the current through a resistor wasfound to be (2.5 0.1) mA and the voltage across the resistor

(5.5 0.3) V. Determine the resistance of the resistor Rand the

uncertainty UR !

1020.2A102.5

V5.5 33-I

V

R

1042.2A102.4

V8.5 33-max

R

1000.2A102.6

V2.5 33-min

R

10210.02

3minmax RR

102.02.2 3R


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Combining Uncertainty Partial

Differentiation

Based on differentiation of

function of several variables

b

b

Va

a

VV

baVV

),(

baV Ub

VU

a

VU

bbVa

aVV

Uncertainty propagation:

Properties:

baVbaV

baVbaV

b

b

a

a

V

VabV

b

b

a

a

V

VbaV

/

Sum:

Difference:

Product:

Quotient:


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Combining Uncertainty Partial

Differentiation

Example :

The temperature of (3.0 0.2) x 10-1kg of water is raised by (5.50.5) oC by heating element placed in the water. Calculate the

amount of heat transferred to the water to cause this

temperature rise !

J1088.4

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

and

J6907)5.5)()(4186(0.3

mcUUcU

mcQcmQ

UQ

Um

QU

mcQ

mQ

mQ

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

is assumed to be constant(neglecting its uncertainty)

J101.19.6 3Q


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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


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Uncertainty in Linear Fitting of X Y Data

Least Square Method in Linear Fitting

22ii

iiii

xxn

yxyxnm

22

2

ii

iiiii

xxn

yxxyxc


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In reality for each value of x, the corresponding y

has some uncertainty

contribute to the uncertainty of m and c


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Uncertainty of m

22

ii

m

xxn

n

Uncertainty of c

22

2

ii

i

m

xxn

x

where

2

2

1

cmxy

n ii


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PART 3

More on Measurement Uncertainty


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General Uncertainty Analysis

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

a function of Jmeasured 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

XUX

r

ivariablemeasuredin theyuncertaint

tcoefficienysensitivitabsolute

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

the same level of confidence


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Nondimensionalized forms:

222

2

2

2

2

2

1

2

1

12

2

...21

j

X

j

jXXr

X

U

Xr

r

X

X

U

Xr

rX

X

U

Xr

rX

rU j

Note :factorionmagnificatyuncertaint

i

i

i UMFX

r

r

X

oncontributipercentageyuncertaint12

22

UPC

rU

X

U

X

r

r

X

r

i

X

j

i i


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Example:

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

Using ideal gas law, how accurately can the density be determinedif the temperature is measured with an uncertainty of 2 oC and the

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


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Uncertainty analysis:

222

2222

2

2

22

22222

zeroisthatassuming

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


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Uncertainty analysis:

%2.1012.0)298/2(01.0

01.0

298

2

2982732522

22

2

UU

pU

T

U

KTKCU

p

T

o

T


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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 = b ias (systemat ic uncerta inty)

P = precis ion (random) u ncerta inty


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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 PP

1

222

Precision (random) uncertainty:

Correlated systematic

uncertainty


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Systematic Uncertainty

Systematic error can be determined and eliminated by

calibration on ly 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 manufacturers 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 transduceras 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


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Random Uncertainty Analysis

Random uncertainty can be determined with various ways

depending on particular experiment:

o Previous experience of others using the same/similartype of apparatus/instrument

o Previous measurement results using the same

apparatus/instrument

o Make repeated measurement

When making repeated measurement, care should be takento 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 errorso Be careful of using a data acquisition system


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Random Uncertainty Analysis

t

Time, t

Y

Failure to determine random

uncertainty due to inappropriate

data acquisition


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Some Detail Approach/Guidelines

Abernethy approach (1970-1980):

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

estimateconfidence99%for

estimateconfidence95%for2/1

22

rrADD

rrRSS

tSBU

tSBU

Coleman and Steele approach (1989 renewed 1998):

o Adapted in AIAA, AGARD, ANSI/ASME222

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


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Some Detail Approach/Guidelines

ISO Guide approach (1993):

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

oUsing a standard uncertaintyo Instead of categorizing uncertainty as systematic and

random, the standard uncertaintyvalues are divided

into type A standard uncertainty and type B standard

uncertainty

o Type A uncertainties are those evaluated by thestatistical 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 Uto report of all NIST

measurement other than those for which Uchas

traditionally been employed

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

than 2 are only to be used for specific application


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References

[1] Montgomery, D.C., & Runger, G.C, Appl ied Statistics and Probabil i ty for Engineers 3rd

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

[2] Harinaldi, Prinsip-pr insip Statistik Untuk Tekni k dan Sains, Erlangga, (2005)

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

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

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

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

[5] Doebelin, E.O, Engineer ing Experimentation, Plann ing Execution, Reporting, McGraw-

Hill Book Co., New York, (1995)

LM01-Aspek Statistik Pengukuran-Analisis Ketidakpastian

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