Introduction to the guide of uncertainty in measurement

http://economie.fgov.be
Introduction to the Guide of
Uncertainty in Measurement
(GUM)
JCGM 100:2008
JCGM 104:2009
Euramet Workshop on: « Uncertainties in thermometric fixed points & SPRT calibrations »
Brussels, 26, 27 and 28 October 2011
Dr M. Maeck
Ir D. Van Reeth

1. What is measurement uncertainty ?
Outline of this
Introduction to GUM
2. Concepts and basic principles
3. Stages of uncertainty evaluation
4. Modelling – The measurement function
5. The calculation (propagation)

What is measurement
uncertainty ?
measurement
uncertainty (of measurement)
expanded uncertainty
coverage factor
combined standard uncertainty
type A evaluation (of uncertainty)
type B evaluation (of uncertainty)

What is measurement
uncertainty ?
uncertainty (of measurement)
parameter, associated with the result of a measurement,
that characterizes the dispersion of the values that
could reasonably be attributed to the measurand
measurement
process of experimentally obtaining one or more
quantity values that can reasonably be attributed
to a quantity (measurand)
expanded uncertainty
quantity defining an interval about the result of a
measurement that may be expected to encompass a large
fraction of the distribution of values that could reasonably
be attributed to the measurand

What is measurement
uncertainty ?
coverage factor
numerical factor used as a multiplier of the combined
standard uncertainty in order to obtain an expanded
uncertainty
type A evaluation (of uncertainty)
method of evaluation of uncertainty by the statistical
analysis of series of observations
combined standard uncertainty
standard uncertainty of the result of a measurement when
that result is obtained from the values of a number
of other quantities, equal to the positive square root of a
sum of terms, the terms being the variances or
covariances of these other quantities weighted according to
how the measurement result varies with changes
in these quantities
type B evaluation (of uncertainty)
method of evaluation of uncertainty by means other than
the statistical analysis of series of observations

Concepts and
basic principles
random variable (variate)
central limit theorem
"true" value, error and uncertainty
type A standard uncertainty evaluation
t-distribution
effective degrees of freedom

xdPCDF
x
 0
).( 
Concepts and
basic principles
random variable (variate)
a variable that may take any of the values of a specified set of
values and with which is associated a probability distribution (PDF)
From "simulation of PDFs.xls" [uniform distribution (0,1) ; 106 realizations]
0
2000
4000
6000
8000
10000
12000
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
0
10
20
30
40
50
60
70
80
90
100
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
PDF
Cumulated distribution
function (CDF)
2
1
2
011
221
0


  dxxxmeanPDF = P(x) = 1   3
1
3
011
221
0
22


  dxxx mean
   
12
1
4
1
3
1222
 xx meanmean

12
1


Concepts and
basic principles
From "simulation of PDFs.xls"
[sum of two uniform distributions (0,1) ; 106 realizations]
  6
7
3
14
4
15
4
1
2)2(
2
1
2
2
1
3
1
0
3
2
1
2
1
0
22
  dxxdxxdxxdxxxdxxxx mean
   
6
1
1
6
7222
 xx meanmean

6
1

0
5000
10000
15000
20000
25000
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0
0
10
20
30
40
50
60
70
80
90
100
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0
Normalization: Striangle = 1 = (2 - 0).h/2 → h = 1
x2/2 (0 x  1)
CDF =
-x2/2 + 2x -1 (1 x  2)
x (0 x  1)
PDF = P(x) =
-x + 2 (1 x  2)

Concepts and
basic principles
From "central limit theorem.xls"
[sum of 12 uniform distributions (0,1) ; 106 realizations]
central limit theorem
If a random variable Y has population mean μ and population
variance σ2, then the sample mean, y, based on n observations,
has an approximate normal distribution with mean μ and variance
σ2/n, for sufficiently large n
(The Cambridge Dictionary of Statistics)
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
50000
0,0 1,2 2,4 3,6 4,8 6,0 7,2 8,4 9,6 10,8 12,0
0
10
20
30
40
50
60
70
80
90
100
0,0 1,2 2,4 3,6 4,8 6,0 7,2 8,4 9,6 10,8 12,0
µ = 6 ;  = 1 → ePDF
x
2
)6( 2
2
1 




Concepts and
basic principles
Type A standard uncertainty evaluation (GUM 4.4)
0
5
10
15
20
25
96 97 98 99 100 101 102 103 104
0
1
2
3
4
5
6
96 97 98 99 100 101 102 103 104
i ti
1 101.11
2 96.90
3 100.07
4 100.95
5 101.84
6 98.61
7 100.33
8 101.57
9 98.18
10 99.89
11 98.25
12 102.72
13 99.56
14 101.20
15 99.49
16 99.74
17 102.36
18 100,68
19 99.03
20 100.42
tm = 100.145 s(ti) = 1.489 s(tm) =s(t)/20 = 0.333 = u(tm)
20 repeated observations ti of the temperature t
N
N
i
i
m
t
t

 1
 
 
1
1
2


 

N
mi
s
N
i
i
tt
t (N = 20)
     
)1(
1
2


 

NN
mi
N
s
s
N
ii
m
ttt
t (N = 20)
fi fi,c
96 0 0.1
97 1 0.6
98 0 1.9
99 3 4.0
100 5 5.3
101 5 4.5
102 4 2.5
103 2 0.9
104 0 0.2
fi fi,c
96 0 0.0
97 1 0.0
98 0 0.0
99 3 0.1
100 5 21.8
101 5 0.9
102 4 0.0
103 2 0.0
104 0 0.0

Concepts and
basic principles
"True" value, error and uncertainty (GUM D.6)
Uncorrected mean
of observations (sample)
xm  u(xm)
uuu correctioncombined xm
22

Corrected mean
of observations (sample)
Correction for all recognized
systematic effects
xm,c  u(xm)
u(correction)
xm,c  ucombined

Concepts and
basic principles
"True" value, error and uncertainty (GUM D.6)
µc+u
Unknown corrected mean
PDF of (normal) population
unknown combined errorunknown "random" error
in the uncorrected mean
of the observations
unknown value
of measurand
unknown error due to an
unrecognized systematic effect
µu-u
Unknown uncorrected mean
PDF of (normal) population
unknown error due to all
recognized systematic effects
µu - µc
is error
xm - µi
is error

Concepts and
basic principles
t-distribution
If z is a normally distributed random variable with expectation μz
and standard deviation σ, and zm is the arithmetic mean of n
independent observations zk of z with s(zm) the experimental
standard deviation of zm, then the distribution of the variable
t = (z−μz)/s(zm) is the t-distribution or Student's distribution with
 = n − 1 degrees of freedom
From "simulation of PDFs.xls"
 = 4 ; µ = 0  = 24 ; µ = 215.36

Concepts and
basic principles
effective degrees of freedom
In general, the t-distribution will not describe the distribution of
the variable (y − Y)/uc(y) if uc
2(y) is the sum of two or more
estimated variance components even if each component is the
estimate of a normally distributed input quantity. However, the
distribution of that variable may be approximated by a t-
distribution with an effective degrees of freedom veff obtained
from the Welch-Satterthwaite formula



 N
i i
i
c
eff
N
i
yu
yu
yuyu
1
4
4
1
22
c
)(
)(
then)()(if
u
u


N
i
ieff
1
with uu

Stages of uncertainty
evaluation
the formulation stage
the calculation stage

evaluation
The formulation stage
Express mathematically the relationship between the measurand Y
and the input quantities Xi on which Y depends: Y = f (X1, X2, ..., XN).
The function f should contain every quantity, including all
corrections and correction factors, that can contribute a significant
component of uncertainty to the result of the measurement.
Determine xi, the estimated value of input quantity Xi, either on
the basis of the statistical analysis ofseries of observations or by
other means.
Evaluate the standard uncertainty u(xi) of each input estimate xi.
Each standard uncertainty is evaluated following Type A
(statistical analysis of series of observations) or Type B (other
means) procedures.
Evaluate the covariances associated with any input estimates
that are correlated.

evaluation
The calculation stage
Calculate the result of the measurement, that is, the estimate y of
the measurand Y, from the functional relationship f using for the
input quantities Xi the estimates xi obtained earlier.
Determine the combined standard uncertainty uc(y) of the
measurement result y from the standard uncertainties and
covariances associated with the input estimates. If the
measurement determines simultaneously more than one output
quantity, calculate their covariances.
If it is necessary to give an expanded uncertainty U, whose
purpose is to provide an interval y − U to y + U that may be
expected to encompass a large fraction of the distribution of
values that could reasonably be attributed to the measurand Y,
multiply the combined standard uncertainty uc(y) by a coverage
factor k, typically in the range 2 to 3, to obtain U = k.uc(y).
Report the result of the measurement y together with its
combined standard uncertainty uc(y) or expanded uncertainty U.
Describe how y and uc(y) or U were obtained.

Modelling – The
measurement function
Resistance measurement of an SPRT in FP and TPW reference cell
Measurand: W = RFP/RTPW
Input quantities: bridge readings (XFP and XTPW)
XFP= RFP/RS1 → RFP = RS1.XFP
XTPW= RTPW/RS2 → RTPW = RS2.XTPW
XRS
XRS
R
RW
TPW
FP
TPW
FP



2
1
reference resistance for FP
reference resistance for TPW
bridge reading à FP
bridge reading à TPW

Modelling – The
bridge
standard resistor
TPW cells
SPRT

Modelling – The
00RS1121111  DrDrRSRS
00RS2222122  DrDrRSRS
Reference resistances can experience two kinds of drift
drift of the resistance itself
drift from temperature variation of the resistance
distributed as rectangular PDF centered on null value
XDrDrRS
XDrDrRS
R
RW
TPW
FP
TPW
FP



)22212(
)12111(
Drift for resistance Drift for temprature

Modelling – The
C1: difference from national realization of FP [N (µ,)]
C2: chemical impurities in FP cell [N (µ,s)]
C5: choice of the points in the plateau [R (µ,a/2)]
C6: reproducibility for different realizations of plateau [R (µ,a/2)]
C7: AC/DC current [R (µ,a/2)]
C8: bridge linearity [N (µ,s)]
C9: differential bridge non-linearities [R (µ,a/2)]
C10: SPRT self-heating [t (µ,u)]
C11: hydrostatic head correction [R (µ,a/2)]
C12: perturbing heat exchanges [R (µ,a/2)]
XFP is affected by 10 possible contributions:
XDrDrRS
CCCCCCCCCCXDrDrRS
W
TPW
FP



)22212(
)1211109876521()12111(

Modelling – The
XTPW is affected by 13 possible contributions:
)13121110987654321()22212(
)1211109876521()12111(
BBBBBBBBBBBBBXDrDrRS
CCCCCCCCCCXDrDrRS
W
TPW
FP



B1: difference from national realization of TPW [R (µ,a/2)]
B2: chemical impurities in TPW cell [R (µ,a/2)]
B5: daily repeatibility [N (µ,s)]
B6: reproductibility for different ice mantles [N (µ,s)]
B7: AC/DC current [R (µ,a/2)]
B8: bridge linearity [N (µ,s)]
B9: differential bridge non-linearities [R (µ,a/2)]
B10: SPRT self-heating [t (µ,u)]
B11: hydrostatic head correction [R (µ,a/2)]
B12: perturbing heat exchanges [R (µ,a/2)]
B3: isotopic composition of TPW cell [R (µ,a/2)]
B4: gas pressure [R (µ,a/2)]
B13: internal insulation leakage [R (µ,a/2)]

The calculation -
propagation
the law of propagation of uncertainties
type B standard uncertainty evaluation for the
volume of a cylinder, independent and correlated
input quantities
uncertainty from a least squares prediction -
calibration of a thermometer (GUM example H.3)
the numerical estimation of sensitivity coefficients

The calculation -
propagation
Law of propagation of uncertainties
From a Taylor expansion around x0:
    ...
2
1
)()()( 0
2
2
2
00 















 xx
x
z
xx
x
z
xzxzXz
 xx
x
z
xz 00)( 







 (first order limited)
   yy
y
z
xx
x
z
x yzyxzYXz 0000 ),(),(),( 
















   yy
y
z
xx
x
z
x yzyxz iii i 















 ),(),(
   yy
y
z
xx
x
z
zz iii 















 (first order limited)
For a series of measurements (xi, yi) around the mean values x, y:

The calculation -
propagation
Law of propagation of uncertainties
             yyixxiy
z
x
z
yyiy
z
xxi
x
z
zzi 





















 2
2
2
2
2
2
              























N
i
N
i
N
i
N
i
yyixxiy
z
x
z
yyiy
z
xxi
x
z
zzi
11
2
2
1
2
2
1
2
2
   yy
y
z
xx
x
z
zz iii 















 (i = 1, 2, … N)
 
)1(
)( 1
2
2




NN
xxi
xs
N
i
 
)1(
)( 1
2
2




NN
zzi
zs
N
i
 
)1(
)( 1
2
2




NN
yyi
ys
N
i
       
)1(
2)()()( 12
2
2
2
2


























NN
yyixxi
y
z
x
z
ys
y
z
xs
x
z
zs
N
i
covariance termcombined variances

The calculation -
propagation
Type B standard uncertainty evaluation
for the volume of a cylinder
h
d
Six measurements of height and diameter:
i 1 2 3 4 5 6
di /cm 4.985 5.000 5.020 4.975 4.980 5.005
hi /cm 5.980 5.975 6.015 6.000 5.985 5.985
d = 4.994 cm
sdi = 0.017 cm
sd = 0.007 cm
h = 5.990 cm
shi = 0.015 cm
sh = 0.006 cm
V = .d2.h/4 = 117.3391 … cm3

The calculation -
propagation
Type B standard uncertainty evaluation for the
volume of a cylinder, independent input quantities
V/h = .d2
/4 → (V/h)2
= 2
.d4
/16 = 383.74 cm4
sensitivity coefficients:
V/d = .d.h/2 → (V/d)2
= 2
.d2
.h2
/4 = 2208.10 cm4
u2
= (V/h)2
.(sh)2
+ (V/d)2
.(sd)2
cm6
combined variances:
u2
= (383.74).(0.006)2
+ (2208.10).(0.007)2
cm6
u2
= 0.014 + 0.108 = 0.122 cm6
combined standard uncertainty:
uc = 0.35 cm3
u2(h) = 11.5%
u2(d) = 88.5%

The calculation -
propagation
volume of a cylinder, correlated input quantities
u2
= 0.014 + 0.108 + 0.029 = 0.151 cm6
uc = 0.39 cm3
S(di-d).(hi - h) = 4.75.10-4 cm2 V/h = 19.59 cm2 V/d = 46.99 cm2
combined variances-covariance:
(previously 0.35 cm3)
u2(h) = 9.3 %
u2(d) = 71.5 %
u2(h,d) = 19.2 %
covariance term: 2.(V/h).(V/d).S(di – d).(hi – h)/[N.(N-1)]
i 1 2 3 4 5 6
di /cm 4.985 5.000 5.020 4.975 4.980 5.005
hi /cm 5.980 5.975 6.015 6.000 5.985 5.985
(di-d).(hi-h) /cm2
9.16E-05 -8.75E-05 6.46E-04 -1.92E-04 7.08E-05 -5.42E-05
u2
= 0.014 + 0.108 + 2.(19.59).(46.99).(4.75).10-4
/30

The calculation -
propagation
u2
= 0.014 + 0.108 + 0.074 = 0.196 cm6
uc = 0.44 cm3
S(di-d).(hi - h) = 1.20.10-3 cm2 V/h = 19.59 cm2 V/d = 46.99 cm2
(previously 0.39 cm3)
u2(h) = 7.1 %
u2(d) = 55.1 %
u2(h,d) = 37.8 %
i 1 2 3 4 5 6
di /cm 4.985 5.000 5.020 4.975 4.980 5.005
hi /cm 5.980 5.975 6.015 6.000 5.985 5.985
i 1 2 3 4 5 6
di /cm 4.975 4.980 4.985 5.000 5.005 5.020
hi /cm 5.975 5.980 5.985 5.985 6.000 6.015

The calculation -
propagation
i 1 2 3 4 5 6
di /cm 4.985 5.000 5.020 4.975 4.980 5.005
hi /cm 5.980 5.975 6.015 6.000 5.985 5.985
i 1 2 3 4 5 6
di /cm 4.975 4.980 4.985 5.000 5.005 5.020
hi /cm 5.975 5.980 5.985 5.985 6.000 6.015
R2
= 0,1395
5.970
5.980
5.990
6.000
6.010
6.020
4.970 4.980 4.990 5.000 5.010 5.020 5.030
di /cm
hi/cm
R2
= 0,8900
5.970
5.980
5.990
6.000
6.010
6.020
4.970 4.980 4.990 5.000 5.010 5.020 5.030
di /cm
hi/cm
r = 0.373 r = 0.943

The calculation -
propagation
a useful relation:
)()(
)1(
)(
)1(
)(
.
)1(
)()(
22
hsdsr
NN
hhi
NN
ddir
NN
hhdd ii








 
covariance term = )()(2 hsds
h
V
d
V
r 






 0 0 or  0
)1(
)()(
2









NN
hhdd
h
V
d
V ii
covariance term =
definition of r =
 



)()(
)()(
22
hhiddi
hhdd ii
(correlation coefficient)

The calculation -
propagation
covariance can reduce uncertainty:
r = - 0.865
s(d) = 0.007 cm
s(h) = 0.006 cm
i 1 2 3 4 5 6
di /cm 4.975 4.980 4.985 5.000 5.005 5.020
hi /cm 6.015 6.000 5.985 5.985 5.980 5.975
V/h = 19.59 cm2
V/d = 46.99 cm2
uc = 0.23 cm3
(0.35 cm3 for independant input quantities)
(increase)
(decrease)
u2
= 0.014 + 0.108 + 2.(-0.865).(46.99).(19.59).(0.007).(0.006) = 0.055 cm6

The calculation -
propagation
The numerical estimation of sensitivity coefficients
x
xzxxz
x
xzxxz
x
z
xzz
x 














)()()()(
lim)(
0
following a proposal from Kragten [Analyst, 119, 2161-2166 (1994)]
x = sx not really small but convenient
    )(
)(
xzsxzs
x
z
s
xzsxz
x
z
xx
x
x


















contribution of x to uncertainty
three cells in a spreadsheet

The calculation -
propagation
From "spreadsheet evaluation of uncertainty.xls"
V = .d2.h/4
v(d,h) v(d+sd, h) v(d, h+sh)
(V/d).sd (V/h).sh
d , sd , d+sd h , sh , h+sh
(V/d)2.s2
d
(V/h)2.s2
h
uc(V)
(covariance term in E10 cell)

The calculation -
propagation
r(d,h) = 0.000 r(d,h) = 0.373
(exact: 0.35) (exact: 0.39)

The calculation -
propagation
r(d,h) = 0.943 r(d,h) = -0.865
(exact: 0.23)(exact: 0.44)

The calculation -
propagation
Formulas for linear least squares regression: y = a0 + a1.x
Hypothesis: negligible uncertainty on xi reading values → s(xi) = 0
ts)measuremenn;n...1(i//   nyynxx ii
    )()()()(
22
yyxxSyyiSxxiS iixyyyxx
(means)
(sums of squares)
Preliminary calculations
xaayxayaSSa kxxxy  10k101 ˆ
Model parameters
  s
n
SaS
n
yiyi
s y
xyyy
xy i
21
2
2
.
22
ˆ






  (residual variance)
Model performance
S
x
n
s
S
s
xx
xy
xx
xy
2
.a
.
a
1
ss 01

Uncertainties on model parameters

The calculation -
propagation
What about the uncertainty on a predicted value ( y ) ?^
xaay  10ˆ
    00
ˆˆ
),(20
ˆˆ
1010
10
10
2
1
2
2
0
2
2
ˆ 





















 ss
a
y
a
y
aars
a
y
s
a
y
s aaaay
the hard way
xaya  10  xaxayy  11ˆaxxyy 1)(ˆ 
    S
s
xx
n
s
sxxss
a
y
s
y
y
s
xx
xyxy
ayayy
2
.2
2
.2222
1
2
2
2
2
ˆ )()(
ˆˆ
11







S
xx
n
ss
S
xx
n
ss
xx
xyy
xx
sy xy
)(1)(1 2
.ˆ
2
22
ˆ .






 

demonstration: GUM H 3.5

The calculation -
propagation
Calibration of a thermometer (GUM H.3)
This example illustrates the use of the method of least
squares to obtain a linear calibration curve and how
the parameters of the fit, the intercept and slope, and
their estimated variances and covariance, are used to
obtain from the curve the value and standard
uncertainty of a predicted correction.
A thermometer is calibrated by comparing n = 11 temperature
readings tk of the thermometer, each having negligible uncertainty,
with corresponding known reference temperatures tR, k in the
temperature range 21 °C to 27 °C to obtain the corrections
bk = tR, k − tk to the readings. The measured corrections bk and
measured temperatures tk are the input quantities of the evaluation.
b(t) = y1 + y2.(t – t0) (t0 = 20 °C)

The calculation -
propagation
Calibration of a thermometer (GUM H.3)
k tk (°C) tk-20 (°C) bk (°C)
1 21,521 1,521 -0,171
2 22,012 2,012 -0,169
3 22,512 2,512 -0,166
4 23,003 3,003 -0,159
5 23,507 3,507 -0,164
6 23,999 3,999 -0,165
7 24,513 4,513 -0,156
8 25,002 5,002 -0,157
9 25,503 5,503 -0,159
10 26,010 6,010 -0,161
11 26,511 6,511 -0,160
r2
= 0,5427
-0,173
-0,171
-0,169
-0,167
-0,165
-0,163
-0,161
-0,159
-0,157
-0,155
1 2 3 4 5 6 7
b(tk) (°C) bk - b(tk) (°C)
-0,1679 -0,0031
-0,1668 -0,0022
-0,1657 -0,0003
-0,1646 0,0056
-0,1635 -0,0005
-0,1625 -0,0025
-0,1614 0,0054
-0,1603 0,0033
-0,1592 0,0002
-0,1581 -0,0029
-0,1570 -0,0030
y1 = -0.1712 °C
s(y1) = 0.0029 °C
y2 = 0.00218
s(y2) = 0.00067
From file “calibration of a thermometer t0 = 20.xls"
b(30°C) = -0.1712 + 0.00218.(30-20) = -0.1494 °C
s[b(30°C)] = 0.003498.[(1/11) + (10 – 4.008)2/27.419]0.5
= 0.0041 °C (u = 11 – 2 = 9)

true and precise
not true, precise
true, not precise
not true, not precise

Introduction to the guide of uncertainty in measurement

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Introduction to the guide of uncertainty in measurement