2145-391 Aerospace Engineering Laboratory I

1

2145-391 Aerospace Engineering Laboratory I Measurement

Measurement Errors / Models

Measurement Problem and The Corresponding Measurement Model

Measure with Single Instrument: Single Sample / Multiple Samples

Measure with Multiple Instruments: Single Sample / Multiple Samples

Uncertainty of A Measured Quantity (VS Uncertainty of A Derived Quantity – Next Week)

Measurement Statement

Single Sample Measurement

Multiple Samples Measurement

22 ],[,],[, EEXX iiTruei

limit confidence % @ CUXX Xi

limit confidence % @ CUXX X

2

Some Details of Contents

Where are we?

Measured Quantity VS Derived Quantity

Objectives and Motivation

Deterministic Phenomena VS Random Phenomena

Measurement Problems, Measurement Errors, Measurement Models

Population and Probability

Probability Distribution Function (PDF)

Probability Density Function (pdf)

Expected Value

Moments

Sample and Statistics

Sample Mean and Sample Variance

Interval Estimation

Terminologies for Measurement: Bias, Precise, Accurate

Error VS Uncertainty


Measured Variable as A Random Variable

Uncertainty of A Measured Quantity [VS Uncertainty of A Derived Quantity – Next Week]


Experimental Program: Test VS Sample

Some Uses of Uncertainty

4

Where are we on DRD?

bottommost level

bottommost level

Week 3: Instruments

Week 4: Measurement and Measurement Statement:%@ CUX X

Week 1: Knowledge and Logic

Week 2:

• Structure and Definition of an Experiment

• DRD/DRE

5

Measured Quantity VS Derived Quantity

Recall the difference between

Measured Quantity y

(numerical value of y is determined

from measurement with an instrument)

In this period we focus first on measured quantity.

Derived Quantity y

(numerical value of y is determined

from a functional relation)

VS

6

Objectives

limit confidence % @ CUXX XateBest Estim

If you

know what this measurement statement says (regarding measurement result),

know its use [what good is if for, and when to use it]

know and understand its underlying ideas [why should we report a measurement result with this

statement],

(know, understand, and) know how to report a measurement result of a measured quantity with

this measurement statement for the cases of

A single-sample measurement

A multiple-sample measurement

we can all go home.

Activity: Class Discussion on the above

Measurement Statement:

limit confidence % @ CUXX Xi

limit confidence % @ CUXX X

7

Class Activity and Discussion

10 students come up and measure the resistance of a given resistor.

Then, report the measurement result.

Discussions

Do they get the same result?

If not, what is, and who gets, the ‘correct’ value then?

Wanna bet 500 bahts? [on whom, or which value]

If the next 10 of your friends come up and measure the resistance, and

less than 9 out of 10 of them do not get the value you bet on,

you lose and give me 500 bahts. Otherwise, I win.

Or

I’ll let you set the term of our bet. [Of course, I have to agree on the term first.]

What term should our bet be then? [Make it reasonable and “bettable.”]

00000000000000010000000000 RRight. I can just simply close my eyes and bet against

8

Motivation

Given that there are some (random) variations in repeated measurements,

how should we report a measurement result so that it makes some usable

sense?

limit confidence % @ CUXX XateBest Estim

The Why

The Use

The How

of Measurement Statement

9

Deterministic Phenomena VS Random Phenomena

Deterministic Variable VS Random Variable

10

Deterministic Phenomena Determinsitic Variable

Deterministic Phenomena

(1) The state of the system at time t, and

(2) its governing relation,

deterministically determine the state of the system – or the value of the deterministic variable y -

at any later time.

Example: Free fall (and Newton’s Second law)

2/)(:ofvalueFuture

0)0(,0)0(:IC

,:GE

2

2

2

gttyy

yy

gdt

yd

In reality, however, chances are that we will not have that exact position )5( sty

11

Random Phenomena [Random or Statistical Experiment]Random Variable

A random or statistical experiment is an experiment in which [1]

1. all outcomes of the experiment are known in advance,

2. any realization (or trial) of the experiment results in an outcome that is not known in

advance, and

3. the experiment can be repeated under nominally identical condition.

[1] Rohatgi, V. K., 1976, An Introduction to Probability Theory and Mathematical Statistics, Wiley, New York, p. 20.

12

Class Activity: Discussion

Is tossing a coin a random experiment?

1. All possible outcomes are H and T and nothing more.

2. For any toss, we cannot know/predict the outcome in advance.

H or T?

3. Care can be taken to repeat the toss under nominally identical condition.

0)(,1)(},{ TyHyTH

13

Measurement:

1. All possible outcomes are known in advance, e.g.,

2. For any one realization , we cannot know/predict the exact outcome with

certainty in advance.

3. Care can be taken to repeat the measurement under nominally identical

condition.

Class Activity: Discussion

Is measurement a random experiment?

itself)R(notRresistanceMeasured, yy-

......,97.4)2(,99.4)1( yy

14

Measurement Problems

Measurement Errors

Measurement Models

15

Measurement Problems,Measurement Errors,and The Corresponding Measurement Models

Measure with a single instrument

Single Sample

Multiple Samples

Measure with multiple instruments

Single Sample

Multiple Samples

Other models are possible, depending upon the nature of errors considered.

variable.randomais,variablerandoma,constantais:

][,0][,][,0][,:1 2222

not

EEXXModel

i

iTruei

variables.randomareand:

,,0][][,:2 22

EEXXModel

i

iiTruei

16

Measurement Problem: Measure with A Single Instrument: Error at Measurement i

TrueX

i

X

Error at measurement i

iX

Measurement iTrueX

Trueii XX :Total measurement error for reading i :

TrueXTrue value of X :

iXMeasured value of X at reading/sample i :

We never know the true value.

iTruei XX

17

Decomposition of ErrorSystematic/Bias Error VS Random/Precision Error

i

TrueX

i

ii :X

Statistical Experiment:

Repeated measurements under

nominally identical condition

iX

Measurement i

)subscript(with iiRandom/Precision error for reading i,

Randomly varying from one realization to another.

)subscript(no iSystematic/Bias error:

Constant. Does not change with realization.

Measurement model iTruei XX

18

Measurement Problem: Measure with A Single Instrument: Measurement Model 1

The ith measured value

The ith observed value of a random variable X

iX

True value

[Can never be known for certainty, not a random variable]

TrueX

Systematic / Bias error

[Constant. Does not change with realization, not a random variable]

Random / Precision error

[Randomly varying from one realization to another, a random variable]

i

Measurement Model 1:

variable.randomais.variablerandoma,constantais:

][,0][,][,0][: 2222

not

EE

i

iTrueXiX

19

Measured Value [Random Variable] How to Describe/Quantify A Random Variable: Distribution of Measured Value [Random Variable]

TrueX X

Measurement i

Statistical Experiment:

Repeated measurements under

nominally identical condition

Freq

uenc

y of

occ

urre

nces

the population distribution

of measured value X

Repeated measurements under nominally identical condition

Description of Deterministic Variable:

State the numerical value of the variable under that condition.

Description of Random Variable

Since it randomly varies from one realization to the next

[even under the same nominal condition and we cannot predict its value exactly in advance],

a meaningful way to describe it is by describing its probability (distribution).

iX

VS 5TrueX

)(, xfXpdf X

20

Probability and Statistics

Statistics – Inductive reasoningGiven properties of a sample, extract information regarding the population.

Probability – Deductive reasoningGiven properties of population, extract information regarding a sample.

Population

),( XX Sample

),( XSX

21

Population and Probability

Probability – Deductive ReasoningGiven properties of population, extract information regarding a sample.

Population

),( XX Sample

),( XSX

22

4 2 0 2 40

0.2

0.4

0.6

0.8

1

trace 1trace 2

fX x( )

FX x( )

x

Probability Distribution Function (PDF)

]],([][:)( xPxXPxFX

xX )(xFXThe probability of an event is the value of

)(xFX

Some Properties

)(][

1)(

0)(

xFxXP

F

F

X

X

X

)(][ xFxXP X

][Event xX

x

23

Probability Density Function (pdf))(xf X

x

XX dttfxF )(:)(dx

xdFxf X

X)(

:)(

x

XX dttfxFxXP )()(][

or

= Area under the pdf curve from to x.

FX x( ) pnorm x 0 1( )

4 2 0 2 40

0.2

0.4

0.6

0.8

trace 1trace 2

fX x( )

FX x( )

x

Some Properties

x

Area

][Event xX

],()(

)()(][

xintfunderArea

xFdttfxXP

X

X

x

X

1)(

dttf X

Thin / high VS Wide / low

24

5 0 50

0.5

1

trace 1trace 2trace 1trace 2

fX x( )

FX x( )

x

Probability of An Event

The probability of an event is

1. the area under the fX(x) curve from to x,

2. the value of FX(x).

xX

PDF FX(x)

pdf fX(x)

)(xFX

x

x

][Event xX

x

X dttfxXP )(][

][ xX

25

5 0 50

0.5

1

trace 1trace 2trace 1trace 2

fX x( )

FX x( )

x

Probability of An Event

The probability of an event is

1. the area under the fX(x) curve from x1 to x2,

2. the value of [FX(x2) - FX(x1)].

21 xXx

PDF FX(x)

pdf fX(x) )()( 12 xFxF XX

x2

x

2

1

)(][ 21

x

x

X dttfxXxP

x1

FX(x2)

FX(x1)21 xXx

Area

FX(x2) - FX(x1)

][ 21 xXx

26

3 2 1 0 1 2 30

0.25

0.5

0.75

1

dunif x 1 1 punif x 1 1 dunif x 2 2 punif x 2 2

xx

Example of Some pdf: Uniform Density Function

otherwise.

0

1

),;(

bxaab

baxUFunction U(x) with parameters a and b:

PDF: U(x;-2,2)PDF: U(x;-1,1)

U(x;-1,1)

U(x;-2,2)

Thin / high VS Wide / low

27

Example of Some pdf: Normal Density Function

Function N(x) with parameters and 2:2

2

2

)(2

2

1),;(

x

exN

N(x;0,1)

x5 0 5

0

0.5

1

dnorm x 0 1 pnorm x 0 1 dnorm x 0 2 pnorm x 0 2

x

N(x;0,2)

PDF: N(x;0,1) PDF: N(x;0,2)

28

Example of Some pdf: Student’s t Density Function

Function t(x) with parameter :

2/)1(2 /1

1

)2/(

2/)1();(

x

xt

t(x; 5)

x5 0 5

0

0.5

1

dt x 5 pt x 5 dt x 20 pt x 20

x

t(x; 20)

PDF: t(x; 5)PDF: t(x; 20)

29

Example of Some pdf: Chi-Squared Density Function

Function (x) with parameter :

0;0

0;)2/(2

1

);(2/1)2/(

2/2

x

xexx

x

x0 20 40

0

0.1

dchisq x 5 dchisq x 10 dchisq x 15 dchisq x 20

x

2(x; 5)

2(x; 10)2(x; 15)

2(x; 20)

30

Expected Values of A Random Variable

Definition: Expected Value of A Random Variable X

The expected value (or the mathematical expectation or the statistical average) of a continuous

random variable X with a pdf fX(x) is defined as

Definition: Expected Value of A Function of A Random Variable X

Let Y = g(X) be a function of a random variable X, then Y is also a random variable,

and we have

However, we can also calculate E(Y) from the knowledge of fX(x) without having to refer to fY(y) as

dxxXxfXE

X

)(:][

dyyyfYE Y )(][

dxxfxgYE X )()(][

31

Moments of A pdf

Definition: Moment About The Origin

The rth-order moment about the origin (of a df) of X, if it exists, is defined as

where r = 0,1,2,….

Note that this is the rth-order moment of area under fX(x) about the origin.

Definition: Central Moment

The rth-order central moment of a df of X, if it exists, is defined as

where r = 0,1,2,….and E[X] = X.

Note that this is the rth-order moment of area under fX(x) about X.

dxxfxXEM Xrr

r )(][ :

dxxfxXEm Xr

Xr

Xr )()(])[( :

32

dx

Interpretations of Moments

NOTE: Due to the rth-power of the arm length, the values of fX (x) at further distance from the

center (origin or X) relatively contribute more to the moment than those at closer to the center.

X

moment arm for Mr = x(r)

moment arm for mr = (x-X)(r)

Area dA = fX(x)dx

x

fX (x)

1

0

)()()(])[( Xr

XXr

Xr

Xr dFxdxxfxXEm

1

0

)(][ Xr

Xrr

r dFxdxxfxXEM

dA

dA

x

x - x

33

Some Properties of Mr and mr

Properties of Origin Moment Mr

Moment order 0:

Moment order 1 (Mean of rv X):

Moment order 2

Properties of Central Moment mr

Moment order 0:

Moment order 1:

Moment order 2:

(Variance X2)

1]1[][ 0

dxfEXEM Xo

dxxfXEM XX][1

dxfxXEM X22

2 ][

1]1[])[( 0

dxfEXEm XXo

0

)()][(1

dxf

dxfdxxfdxfxXEm

XXX

XXXXXX

2222222

2222

][2][)2(

)(])[(

XXXXXX

XXXX

XEXEdxfxx

dxfxXEm

X is the location of the centroid of the pdf.

X2 is a measure of the width of the pdf.

34

Sample and Statistics

Since we do not know the properties of the population ,

we want to estimate them with the statistics drawn from a sample.

Statistics – Inductive reasoning

Population

),( 2XX

Sample

),( 2XSX

),( 2XSX

),( 2XX

Population Mean

Population Variance

X2X 2

XSXSample Mean

Sample Variance

How close is to in some sense?X X

35

Sample Mean and Sample Variance

Definition

Let X1, X2, …, Xn be a random sample from a distribution function fX(x).

Then, the following statistics are defined.

Sample Mean:

Sample Variance:

Sample Standard Deviation:

Sample mean, sample variance, and sample standard deviation are statistics, hence, random

variables, not simply numbers.

n

i

iXn

X1

1

n

iiX XX

nS

1

22 )(1

1

n

iiXX XX

nSS

1

22 )(1

1

Unbiased estimator of X.

Unbiased estimator of .2X

36

Interval Estimation

Assume X is a random variable whose pdf is normal and

Let (X1, X2, …, Xn) be an iid random sample from

Interval Estimation: Probability Distributions of Random Variables

)1,0(~: NX

ZX

X

),(~ 2XXNX

),(~ 2XXNX

)/,(),(~1

: 22 nNNXn

X XXXXn

i

1~/

nX

X tnS

XT

1~

nX

Xi tS

XT

37

Convention on

Area = /2

2/z

2/t

1][P

2/z

2/t


x

pdf

Normal and Students t

Area = /2

1][Pdchisq x 5 dchisq x 10 dchisq x 15 dchisq x 20

x22/1

pdf

22/

Chi Squared

38

Interval Estimation

Theorem 1: Standard Normal Random Variable

If , then .

Z is called a standard normal random variable.

In addition, we have

or

where z/2 denotes the value on the z axis for which /2 of the area

under the z curve lies to the right of z/2.

)1,0(~: NX

ZX

X

),(~ 2XXNX

1 2/z

XZP

X

X 1)( 2/ XX zXP

magnitude of the deviation/distance

from X to X, or from X to X.

39

1)( 2/ XX zXP

The probability that deviates from no more than ( times ) is

.

2/zX X

1

X

40

Theorem 2: Distribution for A Random Variable

Let (X1, X2, …, Xn) be a sample from .

Then, the random variable has

Hence,

or

X

)/,(),(~ 22 nNNX XXXX

1

/ 2/z

n

XZP

X

X

1)/( 2/

nzXP XX

),(~ 2XXNX

X

41

The probability that deviates from no more than ( times )

is .2/

zX X1

nX /

1)/( 2/

nzXP XX

42

Theorem 3: Distribution for A Random Variable

(Student’s t Distribution)

Let (X1, X2, …, Xn) be a sample from .


that is, T has a Student’s t distribution with degree of freedom = n -1.

Hence,

or

nS

XT

X

X

/:

1~/

nX

X tnS

XT

1

/ ,2/t

nS

XTP

X

X

1)/( ,2/

nStXP XX

),(~ 2XXNX

nS

XT

X

X

/:

43

The probability that deviates from no more than ( times )

is . ,2/

tX X1

nS X /

1)/( ,2/

nStXP XX

45

One More Sample from Previously Drawn n Samples (Large Sample Size Approximate, n large)

Let (X1, X2, …, Xn) be a sample from and

be the sample variance of this sample.

Let be an additional single sample drawn from .


that is, T has a Student-t distribution with degree of freedom = n-1.

Hence,

or

X

Xi

S

XT

1~

nX

Xi tS

XT

2XS

iX

1 ,2/t

S

XTP

X

Xi 1)( ,2/ XXi StXP

),(~ 2XXNX

),(~ 2XXNX

46

The probability that deviates from no more than ( times ) is

. ,2/

tiX X1

XS

1)( ,2/ XXi StXP

47

Summary of Interval Estimation Scheme Diagram

48

Interval Estimation

Assume X is a random variable whose pdf is normal and

Let (X1, X2, …, Xn) be an iid random sample from

Interval Estimation: Probability Distributions of Random Variables

)1,0(~: NX

ZX

X

),(~ 2XXNX

),(~ 2XXNX

)/,(),(~1

: 22 nNNXn

X XXXXn

i

1~/

nX

X tnS

XT

1~

nX

Xi tS

XT

2/t

1][P

2/t


x

pdf

Students t

Area = / 2

49

Terminologies for Measurement

Bias

Precise

Accurate

50

Terminologies for Measurement: Bias, Precise, and Accurate

Freq

uenc

y of

occ

urre

nces

XXTrue , X

Unbiased + Precise Accurate

XXTrue, X

Unbiased + Imprecise Inaccurate

Biased + Precise Inaccurate

XXTrue, X

Biased + Imprecise Inaccurate

XXTrue X

51

Error VS Uncertainty

52

Terminologies: Error VS Uncertainty

• Error

If the error is known for certainty, (it is the duty of the experimenter

to) correct it and it is no longer an error.

• Uncertainty

For error that is not known for certainty, no correction scheme is

possible to correct out these errors.

In this respect, the term uncertainty is more suitable.

The two terms sometimes – if not often – are used without strictly adhere to this. Nonetheless, the above should

be recognized.

53


Measured Variable as A Random Variable

Uncertainty of A Measured Quantity

[VS Uncertainty of A Derived Quantity – Next Week]


1001][

limit confidence %@

CUXXUXP

CUXX

XateBest EstimTrueXateBest Estim

XateBest Estim

54

Measurement Model 1: Bias VS Precision/Random (Scatter)

For repeated measurements under nominally identical condition:

Bias: results in the deviation of the (population) mean from the true value.

Precision/Random: results in the scatter in data in a set of repeated measurements.

It is viewed and quantified as

the band width of the scatter

not absolute position.

iTrueX

)( XPWidth

22 ][,0][, EXX iTruei

Area = 1- i

Bias

Population width

XXP

)( XP

)( XPWidth

Sample width

XX SP

)( XP

)( XPWidth

1]|)(|[ XTruei PXXP

X

55

Measurement Model 2 :Replacement Concept: Bias Error as Random Variable

+ B

X: B

ias

Unc

erta

inty

j: One realization of bias error.

XTrue

i

+ P

X : Random

Uncertain

ty

i

One realization of random error

If the measurement instrument identity is changed, the bias is changed and is considered a random variable.

Xj,i = ith realization of instrument j

22 ,,0][][, EEXX iiTruei

56

Uncertainty of A Measured Quantity

XnSt

XSt

P

SS

StB

StU

PBU

StStSt

StStSt

CC

nStXPtnS

XX

CC

StXPtS

XX

SSSSSS

S

XXE

EEXX

Xn

iXn

X

X

nX

XnX

XXX

nnXn

nnXn

U

XnXnX

X

U

XnXinX

Xii

XX

X

TrueX

iiTruei

X

X

Report/samplesMultiple)/(

Report/sampleSingle)(

::

instrumentonewithMeasure:

::

,~

::

,

)()()~

(

~

confidence%@,100

1]/~

||[~/

~Report/SampleMultiple

confidence%@,100

1]~

||[~~Report/SampleSingle

~~samplewithpopulationEstimate:

][

,,0][][,

2/,1

2/,1

2/,1

2/,1

222

22/,1

22/,1

22/,1

222/,1

222/,1

222/,1

2/,11

2/,11

22222

222

22

57

UX = Root-Sum-Square (RSS) of BX and PX

1

22XXX PBU

X

Measurement statement i

%@CUX X

Bias and Precision/Random uncertainties are combined with

root-sum-square (rss) method.

58

Estimating BX and PX

Bias Uncertainty

Bias Uncertainty = Root-sum-square (RSS)

of elemental error sources.

To the very least, it is the uncertainty of the

instrument itself, e.g.,

%@... 222

21 CeeeB MX

%@220, CUUBB IdXX

C%@n)(Resolutio)2/1(0 U

%@...22

21 CeeU I

Precision Uncertainty

Report:

Single value:

Average value:

XnX StP 1,2/

n

St

StP

X

XnX

,2/

1,2/

59

Measurement Statement as An Interval Estimate

22

22

,samplesMultiple:

,sampleSingle:

limit confidence % @

XXXX

XXXXi

XateBest Estim

PBUUXX

PBUUXX

CUXX

100-1

C [P TrueX XateBest Estim UX XateBest Estim UX ]

60

Experimental Program:

Test VS Sample

61

Experimental Program X r = r(X1, …, Xi, …, XJ) rReport value of r

as a final result

Test: The word is associated with the evaluation of DRD/DRE, or r.

One Test = One Evaluation of DRE One r

Measurement, Reading, Sample: The word is associated with Xi reading from the individual

measurement system i.

One Sample = One Measurement of Xi One Xi

Terminologies: Test VS Measurement/Reading/SampleExperiment

),...,,...(:/ 1 Ji XXXrrDREDRD

Single

Multiple

Single

Multiple

Single X Single r

Average X Single r

Average X Single r

Single X Single r

Single r

Single r

Multiple r

Multiple r

Single r

Single r

Average r

Average r

ST/SS

Single Test/Single Sample

ST/MS

Single Test/Multiple Sample

MT/SS

Multiple Test/Single Sample

MT/MS

Multiple Test/Multiple Sample

62

Data Analysis for Various Types of Experiment

Reading/Sample kth Test kth

ST/SS

ST/SS

Average over (Xi)k

ST/MS

ST/MS

MT/MS

Average over rl

MT/SS

Average over rk

11 ),...,,...,(:1 kJi XXXk

kJi XXXkk ),...,,...,(: 1

111 ),...,,...,(: kJik XXXrrr

kJik XXXrrr ),...,,...,(: 1

11 ),...,,...,(:1 lJi XXXl 111 ),...,,...,( lJil XXXrr

lJi XXXll ),...,,...,(: 1 lJil XXXrr ),...,,...,( 1

63

Finally: Summary Measurement Statement and Interval EstimationUncertainty of A Measured Quantity

2/t

1][P

2/t


x

pdf

Students t

Area = / 2

XnX

XXX

U

XnXi

nX

Xi

nX

Xi

Xi

StP

PBU

CStXP

Ct

S

XP

tS

XT

CUXX

X

2/,1

22

2/,1

2/,1

1

:

:

1001]

~[

1001~

~~

limit confidence %@

Single Sample: Report iX

nStStP

PBU

CStXP

Ct

S

XP

tS

XT

CUXX

XnXnX

XXX

U

XnX

nX

X

nX

X

X

X

/:

:

1001]

~[

1001~

~~

limit confidence %@

2/,12/,1

22

2/,1

2/,1

1

Multiple Samples: Report X


1001][:

limit confidence %@

C

XUTrueXiXP

CUXX Xi

Single Sample: Report iX

1001][:

limit confidence %@

C

XUTrueXXP

CUXX X

Multiple Samples: Report X

64

Some Uses of Uncertainty

Interpretation of Experimental Result

Comparing Theory and Experiment

Comparing Two Models

Industry

65

Some Uses of Uncertainty: Interpretation of Experimental Result

x

y

Without uncertainty, we cannot evaluate how good

– or precise - is the current experimental result.

x

y

With uncertainty, at least we have some indications of how good – or precise – is the current experimental result.

x

y

66

Some Uses of Uncertainty: Comparing Theory and Experiment

x

y Model A

Without uncertainty, we cannot compare whether or not the theoretical result is consistent with the experiment result.

x

y Model A

With uncertainty, in this case we see that they are consistent (within the limit of the uncertainty of the current experimental result).

x

y Model A

With uncertainty, in this case we see that they are not consistent (within the limit of the uncertainty of the current experimental result).

Note:

• Often a theoretical result requires constants that must – or should - be determined from experiment.

• As a result, there are uncertainty associated with a theoretical result also.

• Therefore, there should be error bars (though not shown here) associated with theoretical result also.

67

Some Uses of Uncertainty: Comparing Two Models

x

y

Model B

Model A

Without uncertainty of the experimental result,

we cannot differentiate which one, A or B, is better.

With uncertainty of the experimental result,

in this case we see that the performance of

models A and B cannot be differentiated with the

current experimental result.

x

y Model A

Model B

68

Some Uses of Uncertainty: Industry

Would you like to know – roughly:

How accurate (or uncertain) are the flowmeters at gas stations in

Bangkok?

Let’s say, you pay ~ 1,000 bahts/week 52,000 bahts / year.

Is it + 10 % @ 95% CL,

+ 5 % @ 95% CL,

+ 1 % @ 95% CL,

+ 0.5 % @ 95% CL?

• Imagine, e.g., PTT who sells petrol/gas in billions of bahts.

• Since we cannot avoid this uncertainty – but we can try to minimize it, what

would you do if you were, e.g., PTT, and you were uncertain ~ + 10%, + 5%, +

1%, + 0.5% ?

Documents

2145-391 Aerospace Engineering Laboratory I