1
2103-390 ME Exp and Lab 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
Measurement Statement
Measured Variable as A Random Variable
Uncertainty of A Measured Quantity [VS Uncertainty of A Derived Quantity – Next Week]
Measurement Statement
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
dnorm x 0 1 pnorm x 0 1 dnorm x 0 2 pnorm x 0 2
x
Normal and Students t
Area = /2
1][Pdchisq x 5 dchisq x 10 dchisq x 15 dchisq x 20
x22/1
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 .
Then, the random variable has
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 .
Then, the random variable has
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
dnorm x 0 1 pnorm x 0 1 dnorm x 0 2 pnorm x 0 2
x
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
Measurement Statement
Measured Variable as A Random Variable
Uncertainty of A Measured Quantity
[VS Uncertainty of A Derived Quantity – Next Week]
Measurement Statement
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
dnorm x 0 1 pnorm x 0 1 dnorm x 0 2 pnorm x 0 2
x
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
Measurement Statement
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% ?