©1997-2001 by M. Kostic
Ch.5: Uncertainty/Error Analysis
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Introduction
Bias and Precision Errors/Uncertainties
Different Instrument Errors
RSS-Errors Summing
Error Sources
Sensitivity or Dependency Rate
Error Propagation (Combined Uncertainty)
Design-stage Uncertainty
(Instrument errors only)
Advanced-stage Uncertainty
(Instrument and measurement errors)
Multiple- measurement Uncertainty
Error Summation/Propagation
(Expanded Combined Uncertainty)
Problem 5-30
Ch.5: Uncertainty/Error
Analysis
©1997-2001 by M. Kostic
Errors (e = x-x’ ≅ x-xavg= d, also B, P, S≅σ;
do NOT be confused, see NIST Guide):
Bias (B), Precision (P), also Standard (S or σ)
Uncertainty (u) is the range of errors (e, B, P, S)
at corresponding Probability (%P)
Remember: u = d%P = t ν,%PS (@ %P); z=t=d/S
1
Bias and Precision
Errors/Uncertainties
©1997-2001 by M. Kostic
Precision error
Average value
Bias error
True Value
Different Instrument Errors
1
3
2
4
u y = u y1 + u y 2 + ... + u yL =
2
2
©1997-2001 by M. Kostic
2
5
L
∑u
i =1
2
yi
; i.e. RSS ( Root − Sum − Square) procedure
2
©1997-2001 by M. Kostic
RSS-Errors Summing
e1
e2
e3
e4
eSUM=|e 1 |+|e2 |+|e 3 |+|e4 |
eR
e4=Max < e R< eSUM
e1
e3
e2
e4=Max
e1
e R12
eR
2
2
L
∑u
i =1
e4
…more
probable
…or, in general, when eyi is uyi :
u y = u y1 + u y 2 + ... + u yL =
e3
e R123
e R=e RSS = (e1 )2 +(e 2 )2 +(e3 )2 +(e 4 )2
2
…too
conservative
some errors
are opposite
signs and
cancel out
e2
2
yi
; i.e. RSS ( Root − Sum − Square) procedure
©1997-2001 by M. Kostic
Error Sources
1. Calibration Error Source Group
Table 5.1 - (B or P)1j
2. Data Acquisition Error Source Group
Table 5.2 - (B or P)2j
3. Data Reduction Error Source Group
Table 5.3 - (B or P)3j
It is not important which group an error is assigned to,
as long as it is accounted for.
The groups and their items are for convenience only.
3
©1997-2001 by M. Kostic
Sensitivity or Dependency Rate
ity ∂ y
is tiv = ∂x
sen θ y / x
:e
uncertaintyof y
=
p
/x
y
∂
Slo
c
δy ≈ y ux = u y
=
/x
∂x
Ky
δx ≈ ux , uncertainty of x
If one variable " y"depends on another " x, " then a small change of x,
i.e δx ≈ ux (error, uncertainty) will propagate as error of y, i.e. δy ≈ u y .
Using the partial derivative, i.e. the rate of dependancy, or sensitivit y :
δy ≈
∂y
∂y
∂y
δx ≈ ux = (θ y / x )u x = u y ; where sensitivit y K y / x = cy / x = θ y / x =
∂x
∂x
∂x
©1997-2001 by M. Kostic
Error Propagation (Combined Uncertainty)
If one variable y depends on another x, then a small change of
x , i.e δx ≈ u x (error, uncertaint y ) will propagate as error of y ,
i.e. δ y ≈ u y :
δy ≈
∂y
∂y
∂y
δx ≈
u x = (θ y / x )u x = u y ; where sensitivit y θ y / x =
∂x
∂x
∂x
For a multyfunct ion variable y = y ( x1 , x2 ,... xi ,... xL ) :
uy =
∑ (θ
L
i =1
2
y / xi u xi ) =
L
∑u
i =1
2
yi
; i.e. RSS (Root − Sum − Square) procedure
 u yi a known elemental error

 ∂y 
where u yi = 


 or u yi =  ∂x u xi = (θ yi / x ) u xi

i







4
©1997-2001 by M. Kostic
Design-stage Uncertainty
(Instrument errors only)
Design - stage error/uncertainty
u d = u 02 + u c2
Interpolation error Instrument error
u0 = ± 12 resolution uc (by calibration)
Advanced-stage Uncertainty
©1997-2001 by M. Kostic
(Instrument and measurement errors)
N th order uncertainty
N
u N = u + ∑ ui2
2
d
i =1
Zero - and design order uncertaint ies
u0 = ± 1 2 resolution; u d = u02 + u c2
First - order uncertaint y
u1 ≥ u0 , u d
5
Multiple-measurement
Uncertainty
©1997-2001 by M. Kostic
©1997-2001 by M. Kostic
Error Summation/Propagation
(Expanded Combined Uncertainty)
uR =
BR =
∑B
i
B R + ( tν R ,% P PR ) 2 , where :
2
2
i
and
PR =
∑P
i
2
; also :
i
2
νR

2 
 ∑ Pi 
 ; ν = N − 1, # of degree of freedom
= i
i
i
4
Pi
∑i ν
i
Note, it could be Pi = Pyi = θ yi / xi Pxi
or, it could be B i = B yi = θ yi / xi B xi
6
©1997-2001 by M. Kostic
Problem 5-30
©1997-2001 by M. Kostic
Problem 5-30 (Cont.)
7
©1997-2001 by M. Kostic
Problem 5-30 (Cont.)
©1997-2001 by M. Kostic
Problem 5-30 (Cont.)
8
Problem 5-30 (Cont.)
©1997-2001 by M. Kostic
©1997-2001 by M. Kostic
Problem 5-30 (Cont.)
9