Uncertainties of measurement in EXCEL History and basic terms RNDr. Ctibor Henzl, Ph.D. VŠB –...
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Transcript of Uncertainties of measurement in EXCEL History and basic terms RNDr. Ctibor Henzl, Ph.D. VŠB –...
Uncertainties of measurement in EXCEL
History and basic terms
RNDr. Ctibor Henzl, Ph.D.
VŠB – Technical University Ostrava
RNDr. Ctibor Henzl, Ph.D.
VŠB – Technical University Ostrava
Example
References
Conclusions
Uncertainties of measurement represent a statistical approach to the
evaluation of measured data. Alas, approximately since150 years we hear
that there are three forms of falsehoods:
falsehoods
punishable falsehoods
statistics
This often highly cited proposition about statistics is ascribed on Benjamin Disraeli, other times Lord Palmerston is regarded as
the authorBenjamin Disraeli(1804-1881) Lord Palmerston(1784-1865)
In his beautiful book „Moderní statistika, “Dr. Helmut Swoboda indicates two reasons of this abusive statement: 1.Insufficient knowledge of
methods, aims and facilities of the statistics 2.Many people consider statistics
as something which is actually pseudostatistics
The International Committee for The International Committee for Weights and Measures (CIPM) hasWeights and Measures (CIPM) has committed itself since 70 years of committed itself since 70 years of
the last century to create an unique the last century to create an unique methodology for processing and methodology for processing and
evaluating results of measurement evaluating results of measurement
1977 The International Committee for Weights and Measures (CIPM) asked the International Bureau of Weights and Measures (BIPM) to collaborate with national metrology institutes and elaborate recommendations for the solution of the situation, i.e. recommendation of an uniform approach 1980 recommendation is published as
Recommendation INC-1 (1980), see [5].
1990 West European Calibration Committee issued document WECC Doc. 19-1990 see [6] which is a source for a lot of recommendations and norms, see References.
Uncertainties of „A“ type originate from random errors. Their evaluation is based on a statistical analysis of series of observations.
Estimation of uncertainties of „A“ type
The arithmetic mean (the estimate of the quantity) is calculated
mjn
xx
n
i
jij ,...,1,
1
m is number of values, n is number of measurements
The sample variance of is calculated
n
ijjixAx xx
nnsu
jj1
222
1
1
Experimental standard deviation of the mean is used as a standard uncertainty of „A“ type
n
ijjixAx xx
nnsu
jj1
2
1
1
jx
jxs
Uncertainties of „B“ type originate from systematic errors. The evaluation of this uncertainty can not be based on statistical analysis of series of observations. Relevant information are
Technical documentation
Experience with relevant materials and instruments
Knowledge of previous data etc.
Guess zmax – maximal deviation of value, appropriate to source z and look up relevant probability distribution in the following table
Estimation of uncertainties of „B“ type
Determine uncertainty of „B“ type
2
2max2
z
uBz If deviation cannot be practically
exceeded = 3, if deviaton can be exceeded = 2
Gaussian law of propagation of uncertainties
Let us assume that y is a function of values x1, x2, … , xm
),,,( 21 mxxxfy
),,,( 21 mxxxfy
The mean of y is
Uncertainty of the every value x1, x2, … ,xm contributes to the resulting uncertainty
m
xxx
xxx
xxx
mA
x
f
x
fx
f
sss
sss
sss
x
f
x
f
x
fu
mmm
m
m
y
2
1
2
2
2
21
2
2,1,
,221,2
,12,11
,,,
Uncertainty of , symbolized by uAy is calculated by means of Gaussian law of propagation of uncertainties. The “Matrix form“ of this law is very suitable for computer programming.
y
mm xXxX ,,11
The partial derivatives f /xj (referred to as sensitivity coefficients) are evaluated at
There are sample variances of mean of values x1,…,xn in the main diagonal of the matrix
n
ijjiAxx xx
nnus
jj1
222
1
1
m
xxx
xxx
xxx
mA
x
f
x
fx
f
sss
sss
sss
x
f
x
f
x
fu
mmm
m
m
y
2
1
2
2
2
21
2
2,1,
,221,2
,12,11
,,,
kki
n
ijjix xxxx
nns
jk
11
1
There are sample covariance of means in the adjacent diagonal of the matrix
kj xx ,
Off-diagonal elements are zero, if variables x1,…,xn are not correlated. The result is then more simple
m
x
x
x
mA
x
f
x
fx
f
s
s
s
x
f
x
f
x
fu
m
y
2
1
2
2
2
21
2
00
00
00
,,, 2
1
2
2
,,,1
2
1211
j
mm
x
xXxXxX
m
j jAy s
X
fu
Gaussian law of propagation of uncertainties must again be applied when calculating uncertainties of type „B“.
Combined uncertainty
22ByAyCy uuu
Combined uncertainty uCy includes both types of uncertainties. It is computed as the geometrical mean of uAy and uBy
Expanded uncertainty
Expanded uncertainty u is the product of the combined uncertainty uCy and the coverage factor k.
Cyuku
Result
The result can be written in the form
If k = 2, estimated values are approximately normally distributed and expanded uncertainty is uCy, then the unknown value of y is believed to lie in the interval defined by u with a level of confidence of approximately 95%.
uyuyY ;
uyY
Let us compute the Body mass index (BMI) of a group of students.
Example
The personal weighting-machine LUXA and folding rule LOGAREX 38031 were used.
A group of 15 students 15 years old was chosen among the first year students of theTelecommunication school in Ostrava
2l
mBMI
Body mass index is defined as the ratio of mass in kg and the square of height in m.
Mass m and length l are the directly measured values. BMI is calculated for each student..
The arithmetic mean, complete with uncertainty of measurement is calculated for the group.
BMI and health risk are related (see Table).
Very HighObesity 3. level40,0 and more
HighObesity 2. level35,0 – 39,9
AverageObesity 1. level30,0 – 34,9
IncreasedOverweight25,0 – 29,9
MinimalNormal18,5 – 24,9
Health riskStatusBMI
Very HighObesity 3. level40,0 and more
HighObesity 2. level35,0 – 39,9
AverageObesity 1. level30,0 – 34,9
IncreasedOverweight25,0 – 29,9
MinimalNormal18,5 – 24,9
Health riskStatusBMI
n
ijjiAx xx
nnu
j1
22
1
1
2
2max2
z
ujBx
Gaussian law
Table ofresults
Uncertainties of „A“ type are computed using formulas
2
1
22 kg12,21
1
n
immiAm xx
nnu
Commentaries
2
1
22 m0001,01
1
n
illiAl xx
nnu
The formulas are based on statistical analysis of series of observations
CommentariesUncertainties of „B“ type are computed by
2
2max2
z
ujBx
zmax is estimated, is determined from distribution of probability, see table
Commentaries
2
2
2
2
2max2
1max kg3,03
1
3
1rectangle1
z
uon distributikg,z Bm
22
2
2max2
2max kg702,09
5,0 normal50
z
uondistributikg,,z Bm
26
2
2
2
2max2
max m1083
005,0rectangle0050
z
uondistributim,,z Bl
Estimation of „B“ type uncertainties in our example
The partial derivatives f/xi are calculated
from their analytical expression2
22,
21
m32,077,1
11
ll
m
mX
f
llmm
3334
,2
2
mkg2,2477,1
0,67222
l
m
l
lm
l
m
lX
f
llmm
422
2
2
2
2
1
2 mkg28,0
AlAmAy uX
fu
X
fu
Commentaries
We use the Gaussian law of propagation of uncertainties for calculating uncertainties of „A“ type
422
2
2
22
21
2
1
2 mkg04,0
BlBmBmBy uX
fuu
X
fu
The combined uncertainty uc is
222 kgm57,0 yByAc uuu
Commentaries
The Gaussian law of propagation of uncertainties is used for calculating uncertainties of „B“ type
The expanded uncertainty u is
2kgm1,12 cuu
Commentaries
The final result is given by
2-__________
kgm1,14,21 BMI
ConclusionsComputing of uncertainties of A and B
type is described in this paperComputation has been made according
to documents WECC doc. 19-1990 and EAL-4/02
A practical-oriented example has been solved
Spreadsheet EXCEL has been used for computations
References
[1] ISO, Guide to Expression of Uncertainty in Measurement (International Organisation for Standardization), Geneva, Switzerland, 1993
[2] NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, US Government Printing Office, Washington, 1994
[3]Publication Reference EAL-4/02 (European cooperation for Accreditation of Laboratories),1999
[4] WECC Doc. 19-1990 : "Guidelines for Expression of the Uncertainty in Calibrations
[5] http://physics.nist.gov/cuu/index.html
[6] http://www.european-accreditation.org
[7] http://www.bipm.org/CC/documents /JCGM/bibliography_on_uncertainty.html/