Download Measurement and Uncertainty Analysis of Measured

2103-390 ME Exp and Lab I

Measurement

Measurement Errors / Models

X i  X True   i   i ,
E[ ],  2 , E[ ],  2
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
X  Xi U X
@ C % confidence limit

Multiple Samples Measurement
X  X U X
@ C% confidencelimit
1
Some Details of Contents

Where are we?

Measured Quantity

Objectives and Motivation

Deterministic Phenomena VS Random Phenomena

Measurement Problems, Measurement Errors, Measurement Models

Population and Probability

VS
Derived Quantity

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

Measurement Statement
VS
Uncertainty

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
2
Where are we on DRD?
Week 1: Knowledge and Logic
Week 2:
• Structure and Definition of an Experiment
• DRD/DRE
bottommost level
bottommost level
Week 3: Instruments
Week 4: Measurement and Measurement Statement:
X  U X @ C%
4
Measured Quantity
VS
Derived Quantity
Recall the difference between
Measured Quantity y
(numerical value of y is determined
from measurement with an instrument)
VS
Derived Quantity y
(numerical value of y is determined
from a functional relation)
In this period we focus first on measured quantity.
5
Objectives
Measurement Statement:
X  X Best Estimate  U X
@ C % confidence limit
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
X  Xi U X
@ C % confidence limit
 A multiple-sample measurement
X  X U X
@ C% confidencelimit
we can all go home.
Activity: Class Discussion on the above
6
Right. I can just simply close my eyes and bet against 0
Class Activity and Discussion
 R  1000000000000000000000000 
 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.”]
7
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?
The Why
The Use
The How
of Measurement Statement X  X Best Estimate  U X
@ C % confidence limit
8
Deterministic Phenomena
VS
Random Phenomena
Deterministic Variable
VS
Random Variable
9
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)
GE :
IC :
d2y
2
 g,
dt
y (0)  0, y (0)  0
Future value of y :
y (t )  gt 2 / 2
In reality, however, chances are that we will not have that exact position y(t  5 s)
10
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.
11
Class Activity:
Discussion
Is tossing a coin a random experiment?
1. All possible outcomes are H and T and nothing more.
{H , T } 
y( H )  1, y(T )  0
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.
12
Class Activity:
Discussion
Is measurement a random experiment?
Measurement:
1. All possible outcomes are known in advance, e.g.,
-  y   ,
y  Measured resistance R (not R itself)
2. For any one realization w, we cannot know/predict the exact outcome with
certainty in advance.
y(w  1)  4.99 ,
y(w  2)  4.97 ,
......
3. Care can be taken to repeat the measurement under nominally identical
condition.
13
Measurement Problems
Measurement Errors
Measurement Models
14
Measurement Problems,
Measurement Errors,
and The Corresponding Measurement Models

Measure with a single instrument
 Single Sample
 Multiple Samples
Model 1 :
X i  X True     i ,

E[ ]  0,  2 [ ]   2 , E[ ]    0,  2 [ ]   2
i
:  is a constant, not a random variable,  is a random variable.

Measure with multiple instruments
 Single Sample
 Multiple Samples
Model 2 :
X i  X True   i   i ,

E[  ]  E[ ]  0,
 2 ,  2
i
:  and  are random variables.

Other models are possible, depending upon the nature of errors considered.
15
Measurement Problem:
Measure with A Single Instrument: Error at Measurement i
i
Error at measurement i
X True
X True
Measurement i
Xi
X
Measured value of X at reading/sample i :
Xi
True value of X : We never know the true value.
X True
Total measurement error for reading i :
 i : X i  X True
X i  X True   i
16
Decomposition of Error
Systematic/Bias Error VS Random/Precision Error
i
Statistical Experiment:

Repeated measurements under
nominally identical condition
i
Measurement i
X True
 i :    i
Systematic/Bias error:
Xi
X

(no subscript i )
i
(with subscript i)
Constant. Does not change with realization.
Random/Precision error for reading i,
Randomly varying from one realization to another.
Measurement model
X i  X True     i
17
Measurement Problem:
Measure with A Single Instrument: Measurement Model 1
Measurement Model 1:
Xi
Xi




i
i

X True
:
E[ ]  0,  2 [ ]   2 ,
:
 is a constant, not a random variable.  is a random variable.
E[ ]    0,  2 [ ]   2
The ith measured value
The ith observed value of a random variable X
X True
True value
[Can never be known for certainty, not a random variable]

Systematic / Bias error
[Constant. Does not change with realization, not a random variable]
i
Random / Precision error
18
[Randomly varying from one realization to another, a random variable]
Measured Value [Random Variable]
Statistical Experiment:
Repeated measurements under
nominally identical condition
Frequency of occurrences
How to Describe/Quantify A Random Variable: Distribution of Measured Value [Random Variable]
the population distribution
of measured value X
X True
Measurement i
X
Xi
Repeated measurements under nominally identical condition
Description of Deterministic Variable:
VS
State the numerical value of the variable under that condition.
X True  5
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).
19
pdf X , f X ( x)
Probability and Statistics
Probability – Deductive reasoning
Given properties of population,
extract information regarding a sample.
Population
Sample
( X , X )
(X , SX )
Statistics – Inductive reasoning
Given properties of a sample,
extract information regarding the population.
20
Population and Probability
Probability – Deductive Reasoning
Given properties of population,
extract information regarding a sample.
Population
Sample
( X , X )
(X , SX )
21
Probability Distribution Function (PDF) FX (x)
1
P[ X  x]  FX ( x)
fX ( x)
Some Properties
0.8
FX ()  0
0.6
FX ()  1
FX ( x)0.4
P[ X  x]  FX ( x)
Event [ X  x]
0.2
0
4
0x
2
2
4
x
FX ( x) :
trace 1
P [ 2X
trace
 x]  P [ (, x] ]
The probability of an event
X x
is the value of
FX (x)
22
FX ( x)  pnorm
( x0 1) Function (pdf) f X (x)
Probability
Density
Some Properties

0.8
fX ( x)
 f X (t )dt
0.6

Event [ X  x]
Thin / high
VS
Wide / low
x
FX ( x)0.4
P[ X  x] 
 f X (t )dt
 FX ( x )

0.2
0
1
 Area under f X (t ) in (, x]
Area
4
2
0
x 2
4
x
trace 1
x
trace
( x) 2:
FX
 f X (t )dt
or

dFX ( x)
f X ( x) :
dx
x
P[ X  x]  FX ( x) 
 f X (t )dt

= Area under the pdf curve from
 to x.
23
Probability of An Event [ X  x]
1
PDF F
fXX((x)
x)
FX( x)
x
Event [ X  x]
0.5
P[ X  x] 
 f X (t )dt  FX (x)

pdf fX(x)
0
x
5
The probability of an event
0 x
X x
is
1. the area under the fX(x) curve
tracefrom
1
2. the value of FX(x).
5
x

to x,
trace 2
24
Probability of An Event [ x1  X  x2 ]
1
FX(x2)
PDF F
fXX((x)
x)
FX( x)
x2
 f X (t )dt
x1
pdf fX(x)
FX(x1)
0
P[ x1  X  x2 ] 
FX(x2) - FX(x1)
0.5
 FX ( x2 )  FX ( x1 )
Area
x1  X  x2
x
5
x1
0
x2
5
The probability of an event x1  X  x2x is
1 from x1 to x2,
1. the area under the fXtrace
(x) curve
trace 2
2. the value of [FX(x2) - FX(x1)].
25
Example of Some pdf: Uniform Density Function
 1
b  a

U ( x; a, b)  
 0


Function U(x) with parameters a and b:
a xb
otherwise.
1
PDF: U(x;-1,1)


punif x   1  1
0.5
dunif x   2  2
punif x   2  2 0.25
dunif x   1  1 0.75
0
PDF: U(x;-2,2)
Thin / high
VS
Wide / low
U(x;-1,1)
U(x;-2,2)
3
2
1
0
x
x
1
2
3
26
Example of Some pdf: Normal Density Function
Function N(x) with parameters  and 2:
N ( x;  , ) 
1
2
 2

e
( x )2
2 2
1


pnorm x  0  1
dnorm x  0  2
pnorm x  0  2
PDF: N(x;0,1)
dnorm x  0  1
PDF: N(x;0,2)
0.5
N(x;0,1)
N(x;0,2)
0
5
0
xx
5
27
Example of Some pdf: Student’s t Density Function
 (n  1) / 2 
1

 n (n / 2)  1  x 2 /n
Function t(x) with parameter n: t ( x;n )  

n
( 1) / 2
1
 
pt x  5
dt x  20
pt x  20
PDF: t(x; 20)
PDF: t(x; 5)
dt x  5
0.5
t(x; 5)
t(x; 20)
0
5
0
xx
5
28
Example of Some pdf: Chi-Squared Density Function
1

(n / 2 )1  x / 2
x
e
;

n
/
2
2
Function c2(x) with parameter n: c ( x;n )   2 (n / 2)
0;

 
dchisq x  10
dchisq x  15
dchisq x  20
x0
x0
c2(x; 5)
dchisq x  5
c2(x; 10)
c2(x; 15)
c2(x; 20)
0.1
0
0
20
x
40
x
29
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

E [ X ] :  xf X ( x)dx



X
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

E [Y ] 
 yfY ( y)dy

However, we can also calculate E(Y) from the knowledge of fX(x) without having to refer to fY(y) as

E [Y ] 
 g ( x) f X ( x)dx

30
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

M r : E [ X r ] 
r
x
 f X ( x)dx

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

mr : E [( X   X ) ] 
r
r
(
x


)
f X ( x)dx
X


where r = 0,1,2,….and E[X] = X.
Note that this is the rth-order moment of area under fX(x) about X.
31
Interpretations of Moments
moment arm for Mr = x(r)
fX (x)
moment arm for mr = (x-X)(r)
Area dA = fX(x)dx
x
x - x
X
dx

Mr  E [X r ] 



mr  E [( X   X )r ] 


NOTE:
x
1
x r f X ( x)dx   x r dFX
0
dA
1
( x   X )r f X ( x)dx   ( x   X )r dFX
dA
0
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.
32
Some Properties of Mr and mr
Properties of Origin Moment Mr

f X dx  1


M o  E[ X ]  E[1] 
0
Moment order 0:
Moment order 1 (Mean of rv X):
M 1  E[ X ]   X 
Moment order 2
M 2  E[ X ] 

2
Properties of Central Moment mr
Moment order 0:

 xf X dx

2
x f X dx


X is the location of the centroid of the pdf.

mo  E[( X   X ) ]  E[1] 
0
 f X dx  1

Moment order 1:
m1  E[( X   X )] 






 ( x   X ) f X dx   xf X dx    X f X dx

 X  X
 f X dx  0
X2 is a measure of the width of the pdf.

Moment order 2:

m2  E[( X   X )
2
]   X2

 (x   X )
2
f X dx

(Variance X2)


 (x

2
 2 X x   X2 ) f X dx  E[ X 2 ]  2 X2   X2  E[ X 2 ]   X2
33
Sample and Statistics
Population
Sample
( X ,  X2 )
( X , S X2 )
Statistics – Inductive reasoning
Population Mean
X
Sample Mean
Population Variance
 X2
Sample Variance
How close is
X
S X2
X to  X in some sense?
Since we do not know the properties of the population ( X ,  X2 ) ,
we want to estimate them with the statistics ( X , S X2 ) drawn from a sample.
34
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.
n
Sample Mean:
1
X   Xi
n i 1
Unbiased estimator of X.
n
1

( X i  X ) 2 Unbiased estimator of  X2 .

n  1 i 1
Sample Variance:
S X2
Sample Standard Deviation:
SX 
S X2

1 n
( X i  X )2
n  1 i 1

Sample mean, sample variance, and sample standard deviation are statistics, hence, random
variables, not simply numbers.
35
Interval Estimation
Assume X is a random variable whose pdf is normal and
X ~ N ( X , X2 )
Let (X1, X2, …, Xn) be an iid random sample from
X ~ N ( X , X2 )

Interval Estimation: Probability Distributions of Random Variables
Z :
X :
T
X  X
X
1
 Xi
n n
X  X
~
N (0,1)
~
N (  X ,  X2 )  N (  X ,  X2 / n)
~
tn n 1
~
tn  n 1
SX / n
T
Xi  X
SX
36
Convention on a
Normal and Students t
pdf
P[  ]  1  a


pnorm  x 0 1
dnorm x 0 2 Area = a/2
pnorm  x 0 2
dnorm x 0 1
z a / 2
t a / 2
x
za /2
ta /2
Chi Squared
pdf
dchisq x 5
dchisq x 10
dchisq x 15
dchisq x 20
c12a / 2
P[  ]  1  a
Area = a/2
x
ca2 / 2
37
Interval Estimation
Theorem 1:
If
Standard Normal Random Variable
X~
N ( X , X2 ) ,
then
Z :
X  X
X
~ N (0,1) .
Z is called a standard normal random variable.
In addition, we have


X X
P Z 
 za / 2   1  a
X


or


P X   X  ( za / 2 X )  1  a
magnitude of the deviation/distance
from X to X, or from X to X.
where za/2 denotes the value on the z axis for which a/2 of the area under
the z curve lies to the right of za/2.
38


P X   X  ( za / 2 X )  1  a
The probability that
X
deviates from
X
no more than (
z a / 2times  X
) is
1a .
39
Theorem 2:
Distribution for A Random Variable
Let (X1, X2, …, Xn) be a sample from
Then, the random variable
X
X
X ~ N ( X , X2 ) .
has
X ~ N ( X ,  X2 )  N ( X ,  X2 / n)
Hence,


X  X
P Z 
 za / 2   1  a
X / n


or


P X   X  z a / 2 ( X / n )  1  a
40


P X   X  z a / 2 ( X / n )  1  a
The probability that
is
1a
X
deviates from
X
no more than ( z
times
a /2
X / n
.
41
)
Theorem 3:
Distribution for A Random Variable
T :
X  X
SX / n
(Student’s t Distribution)
X ~ N ( X , X2 ).
X   X has
Let (X1, X2, …, Xn) be a sample from
Then, the random variable
T
X  X
SX / n
T :
SX / n
~ tn n 1
that is, T has a Student’s t distribution with degree of freedom n = n -1.
Hence,


X  X
P T 
 ta / 2,n   1  a
SX / n


or


P X   X  t a / 2,n (S X / n )  1  a
42


P X   X  t a / 2,n (S X / n )  1  a
The probability that
is
1a
X
deviates from
X
no more than ( t
times
a / 2 ,n
SX / n
.
43
)
One More Sample from Previously Drawn n Samples (Large Sample Size Approximate, n large)
Let (X1,
S X2
Let
X2, …, Xn) be a sample from X ~ N ( X , X2 ) and
be the sample variance of this sample.
Xi
be an additional single sample drawn from
Then, the random variable
T
T
Xi  X
~ tn  n 1
SX
Xi  X
SX
X ~ N ( X , X2 ) .
has
that is, T has a Student-t distribution with degree of freedom n = n-1.
Hence,


Xi  X
P T 
 ta / 2,n   1  a
SX


or


P X i   X  (t a / 2 ,n S X )  1  a
45


P X i   X  (t a / 2 ,n S X )  1  a
The probability that
1a .
X i deviates from  X
no more than ( t
a / 2 ,n
times
SX )
is
46
Summary of Interval Estimation Scheme Diagram
47
Interval Estimation
Assume X is a random variable whose pdf is normal and
X ~ N ( X , X2 )
Let (X1, X2, …, Xn) be an iid random sample from
X ~ N ( X , X2 )

Interval Estimation: Probability Distributions of Random Variables
Z :
X :
X  X
X
1
 Xi
n n
~
N (0,1)
~
2
2
N ( X , X
)  N ( X , X
/ n)
Students t
T
X  X
pdf
~
SX / n
T
Xi  X
SX
P[  ]  1  a
tn n 1dnorm x 0 1


Area
=a / 2
dnorm x 0 2
tn  n 1pnorm  x 0 2
pnorm x 0 1
~
t a / 2
x
ta /2
48
Terminologies for Measurement
Bias
Precise
Accurate
49
Frequency of occurrences
Terminologies for Measurement: Bias, Precise, and Accurate
XTrue
X
X
Biased + Imprecise  Inaccurate
XTrue, X
X
Unbiased + Imprecise  Inaccurate
XTrue,
X
X
Biased + Precise  Inaccurate
XTrue , X
Unbiased + Precise  Accurate
X
50
Error
VS
Uncertainty
51
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.
52
Measurement Statement
Measured Variable as A Random Variable
Uncertainty of A Measured Quantity
[VS Uncertainty of A Derived Quantity – Next Week]
Measurement Statement
X  X Best Estimate  U X
@ C % confidence limit
P [ X Best Estimate  U X  X True  X Best Estimate  U X ]  1  a
C

100
53
Measurement Model 1: X i  X True     i ,
Bias
VS
E[ ]  0,  2 [ ]   2
Precision/Random (Scatter)
Population width ( PX )
Sample width ( PX )
 PX     X
 PX    S X
i
Bias
Area = 1- a
 Width (  PX )
 Width (  PX )

P [ | X i  ( X True   ) |  PX ]  1  a

X True
X
i
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.
 Width ( PX )
54
Measurement Model 2 : X i  X True   i   i ,
E[  ]  E[ ]  0,  2 ,  2
Replacement Concept: Bias Error as Random Variable
If the measurement instrument identity is changed, the bias is changed and is considered a random variable.
One realization of
random error
i
+ PX: Random Uncertainty
+ BX: Bias Uncertainty
Xj,i = ith realization of instrument j
j: One realization of bias error.
XTrue
i
55
Uncertainty of A Measured Quantity
X i  X True   i   i ,
E[  ]  E[ ]  0,  2 ,  2
E[ X ]   X  X True
 X2   2   2
: Estimate population  with sample S
~
~
S X2  S 2  S2

S X  S 2  S2
Single Sample / Report X i
Multiple Sample / Report X
Xi  X
~ tn  n 1
~
SX

X  X
~ tn  n 1
~
SX / n

~
P [ | X i   X |  tn  n 1,a / 2 S X ]  1  a

 


C
, @ C % confidence
100

C
, @ C % confidence
100
UX
~
P [ | X   X |  tn  n 1,a / 2 S X / n ]  1  a

UX
~
tn2 n 1,a / 2 S X2  tn2 n 1,a / 2 S 2  tn2 n 1,a / 2 S2
~
(tn  n 1,a / 2 S X ) 2  (tn  n 1,a / 2 S  ) 2  (tn  n 1,a / 2 S ) 2
U X2  B X2  PX2 ,


:
~
U X : tn  n 1,a / 2 S X ,
:
B X : tn  n 1,a / 2 S 
:
Measure with one instrument S  S X
:
tn  n 1,a / 2 ( S X )

PX : 
t
 n  n 1,a / 2 ( S X / n )


Single sample / Report X i
Multiple samples / Report X
56
UX = Root-Sum-Square (RSS) of BX and PX
1
X
U X  B X2  PX2
Measurement statement i
X  U X @C %
Bias and Precision/Random uncertainties are combined with
root-sum-square (rss) method.
57
Estimating BX and PX
 Bias Uncertainty
 Precision Uncertainty
2
B X  e12  e22  ...  eM
@ C%
Report:
Bias Uncertainty = Root-sum-square (RSS)
of elemental error sources.
Single value:
PX  ta / 2,n  n 1S X
Average value:
PX  ta / 2,n  n 1S X
To the very least, it is the uncertainty of the
instrument itself, e.g.,
BX  BX ,d 
U 02
 U I2
@ C%
U 0  (1 / 2)(Resolutio n)
@ C%
U I  e12  e22  ...
@ C%
 ta / 2,n
SX
n
58
Measurement Statement
X  X Best Estimate  U X
as
An Interval Estimate
@ C % confidence limit
:
Single sample
X  Xi U X ,
U X  B X2  PX2
:
Multiple samples
X  X U X ,
U X  B X2  PX2
P [ X Best Estimate  U X  X True  X Best Estimate  U X ]  1 - a

C
100
59
Experimental Program:
Test VS Sample
60
Terminologies: Test VS
Experiment
Measurement/Reading/Sample
DRD / DRE :
Test:
r  r ( X 1 ,... X i ,..., X J )
The word is associated with the evaluation of DRD/DRE, or
One Test = One Evaluation of DRE
r.

One r
Measurement, Reading, Sample: The word is associated with Xi reading from the individual
measurement system i.
One Sample = One Measurement of Xi
Experimental Program
ST/SS
Single
Single X  Single r
Single r
Single r
Multiple
Average X  Single r
Single r
Single r
Single
Single X  Single r
Multiple r
Average r
Multiple
Average X  Single r
Multiple r
Average r
Multiple Test/Single Sample
MT/MS
Multiple Test/Multiple Sample
Report value of r
r = r(X1, …, Xi, …, XJ)
Single Test/Multiple Sample
MT/SS
One Xi
X
Single Test/Single Sample
ST/MS

r
as a final result
61
Data Analysis for Various Types of Experiment
Reading/Sample kth
Test kth
Average over rk
k  1 : ( X 1 ,..., X i ,..., X J ) k 1
rk 1 : r  r ( X 1 ,..., X i ,..., X J ) k 1
ST/SS
MT/SS
k  k : ( X 1 ,..., X i ,..., X J ) k
rk : r  r ( X 1 ,..., X i ,..., X J ) k
ST/SS
Average over (Xi)k
l  1 : ( X 1 ,..., X i ,..., X J ) l 1
rl 1  r ( X 1 ,..., X i ,..., X J ) l 1
Average over rl
ST/MS
MT/MS
l  l : ( X 1 ,..., X i ,..., X J ) l
rl  r ( X 1 ,..., X i ,..., X J ) l
ST/MS
62
Finally: Summary
Measurement Statement and Interval Estimation
Uncertainty of A Measured Quantity
Measurement Statement
Students t
pdf
P[  ]  1  a


pnorm  x 0 1
Area = a / 2
dnorm x 0 2
pnorm  x 0 2
dnorm x 0 1
Single Sample: Report
X  Xi U X
Xi
@ C % confidence limit
Multiple Samples: Report
X  X U X
x
t a / 2
ta /2
:
Single Sample: Report X i
Multiple Samples:
X  Xi U X
T
Xi  X
~
SX
U X ]  1  a 
C
100
Report X
~
T
U X
100
X
P [ X  X True
X  X U X
 X 

C
P  i ~ X  tn  n 1,a / 2   1  a 
100
SX


~
C
P [ X i   X  tn  n 1,a / 2 S X ]  1  a 

 

100
C
@ C % confidence limit
@ C % confidence limit
tn  n 1
U X ]  1  a 
P [ X i  X True
:
@ C % confidence limit
X  X
~
SX
~
tn  n 1
 X 

C
X
P
 tn  n 1,a / 2   1  a 
~
100
SX


~
C
P [ X   X  tn  n 1,a / 2 S X ]  1  a 

 

100
U X
:
U X  B X2  PX2
:
:
PX  tn  n 1,a / 2 S X
U X  B X2  PX2
:
PX  tn  n 1,a / 2 S X  tn  n 1,a / 2 S X / n
63
Some Uses of Uncertainty

Interpretation of Experimental Result

Comparing Theory and Experiment

Comparing Two Models

Industry
64
Some Uses of Uncertainty: Interpretation of Experimental Result
y
y
x
With uncertainty, at least we have some indications of how
good – or precise – is the current experimental result.
y
x
Without uncertainty, we cannot evaluate how good
– or precise - is the current experimental result.
x
65
Some Uses of Uncertainty: Comparing Theory and Experiment
y
y
Model A
Model A
x
With uncertainty, in this case we see that they are
consistent (within the limit of the uncertainty of the
current experimental result).
y
Model A
x
Without uncertainty, we cannot compare whether
or not the theoretical result is consistent with the
experiment 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.
With uncertainty, in this case we see that they are not
consistent (within the limit of the uncertainty of the current
experimental result).
x
66
Some Uses of Uncertainty: Comparing Two Models
y
Model A
Model B
x
y
Model A
Model B
x
Without uncertainty of the experimental result,
With uncertainty of the experimental result,
we cannot differentiate which one, A or B, is better.
in this case we see that the performance of models
A and B cannot be differentiated with the current
experimental result.
67
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
Is it
+ 10 %
@ 95% CL,
+5%
@ 95% CL,
+1 %
@ 95% CL,
+ 0.5 %
@ 95% CL?

52,000 bahts / year.
• 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% ?
+
68