Uncertainties of measurement
in EXCEL
History and basic terms
Example
Conclusions
References
RNDr. Ctibor Henzl, Ph.D.
VŠB – Technical University Ostrava
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 author
Benjamin 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
Weights and Measures (CIPM) has
committed itself since 70 years of
the last century to create an unique
methodology for processing and
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
n
x ji
i 1
n
xj  
, j  1,...,m
m is number of values, n is number of
measurements
The sample variance of x j is calculated
2
u Ax
j
n
1
2
2
x ji  x j 
 sx j 

nn  1 i 1
Experimental standard deviation sx j of the
mean is used as a standard uncertainty of „A“
type
u Ax j
n
1
2
x ji  x j 
 sx j 

nn  1 i 1
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
Experience with relevant materials and
instruments
Technical documentation
Knowledge of previous data etc.
Estimation of uncertainties of „B“ type
Guess zmax – maximal deviation of value, appropriate
to source z and look up relevant probability distribution
in the following table
Determine uncertainty of „B“ type
u 
2
Bz
z
2
max
2

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
y  f ( x1 , x2 ,, xm )
Uncertainty of the every value x1, x2, … ,xm
contributes to the resulting uncertainty
The mean of y is
y  f ( x1, x2 ,, xm )
Uncertainty of y , 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.
u
2
Ay
 f
 
,
 x1
f
, ,
x2
 s x21

f  s x2 ,1

xm  
 sx
 m ,1
s x1, 2
s x22

s xm , 2
 f 


 s x1,m  x1 

 s x2 ,m  f 
 x2 
 

 
2 
 s xm  f 
 x 
 m
The partial derivatives f /xj (referred to as
sensitivity coefficients) are evaluated at
X1  x1,, X m  xm
There are sample variances of mean of
values x1,…,xn in the main diagonal of the
matrix
2
sx2j  u Ax
j
n
1
2
x ji  x j 


nn  1 i 1
There are sample covariance of means x j , xk
in the adjacent diagonal of the matrix
sx jk
n
1
x ji  x j xki  xk 


nn  1 i 1
 s x21 s x1, 2

2
s
s

 f
f
f  x2 ,1
x2
2

u Ay   ,
, ,

xm  
 x1 x2
 sx
 m ,1 s xm , 2
 f 


 s x1,m  x1 

 s x2 ,m  f 
 x2 
 

 
2 
 s xm  f 
 x 
 m
Off-diagonal elements are zero, if variables
x1,…,xn are not correlated. The result is then more
 f 
simple


u
2
Ay
 f
 
,
 x1
f
, ,
x2
u
0
s x22

0
0  x1 

 0  f 
 x2 
 

 
2 
 s xm  f 
 x 
 m
2

 f 

 
sx j



X
j 1 
j  X  x , X  x ,, X  x
1
1
2
1
m
m
m
2
Ay
2
 s x21

f  0

xm  
0

Gaussian law of propagation of uncertainties
must again be applied when calculating
uncertainties of type „B“.
Combined uncertainty
Combined uncertainty uCy includes both
types of uncertainties. It is computed as the
geometrical mean of uAy and uBy
uCy  u  u
2
Ay
2
By
Expanded uncertainty
Expanded uncertainty u is the product of
the combined uncertainty
uCy and the
coverage factor k.
u  k  uCy
Result
The result can be written in the form
Y  y u
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%.
Y  y  u; y  u
Example
Let us compute the Body mass index (BMI) of
a group of students.
A group of 15 students 15 years old was
chosen among the first year students of the
Telecommunication school in Ostrava
The personal weighting-machine LUXA and
folding rule LOGAREX 38031 were used.
Body mass index is defined as the ratio of
mass in kg and the square of height in m.
m
BMI  2
l
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).
BMI
Status
Health risk
18,5 – 24,9
Normal
Minimal
25,0 – 29,9
Overweight
Increased
30,0 – 34,9
Obesity 1. level
Average
35,0 – 39,9
Obesity 2. level
High
40,0 and more
Obesity 3. level
Very High
Table
of
results
2
u Ax

j
n
1
x ji  x j 2

nn  1 i 1
u
2
Bx j

2
zmax
2
Gaussian law
Commentaries
Uncertainties of „A“ type are computed using
formulas
u
2
Am
n
1
2
2



x

x

2
,
12
kg

mi
m
nn  1 i 1
n
1
2
2
2


u Al 
x

x

0
,
0001
m

li
l


n n  1 i 1
The formulas are based on
analysis of series of observations
statistical
Commentaries
Uncertainties of „B“ type are computed by
2
z
2
uBx j  max
2

zmax is estimated,  is determined from
distribution of probability, see table
Commentaries
Estimation of „B“ type uncertainties
in our example
z max  1 kg, rectangle distributi on  u
2
Bm1

2
z max

2
zmax  0,5kg, normaldistribution  uBm
2 
2
z max  0,005 m, rectangle distributi on  u Bl

2

2
zmax
2
2
z max
2
12
 3
2
1
  0, 3 kg 2
3
0,52

 0,027 kg2
9

0,005 2
 3
2
 8 10 6 m 2
Commentaries
The partial derivatives f/xi are calculated
from their analytical expression
f
  m
1
1
2

 2 

0
,
32
m
 2
X1 m  l m  m ,l l  l  1,772
 2l   2m  2  67,0
f
  m
3
  2
 m   4   3 


24
,
2
kg

m
3
X 2 l  l m  m ,l l
l
l
1
,
77


We use the Gaussian law of propagation of
uncertainties for calculating uncertainties of „A“
type
2
u
2
Ay
2
 f  2  f  2
  u Am  
  u Al  0,28kg2  m  4
 
 X 1 
 X 2 
Commentaries
The Gaussian law of propagation of
uncertainties is used for calculating uncertainties
of „B“ type
2
u
2
By
2
 f 
 f  2
2
2
  u Bm1  u Bm2  
  u Bl  0,04kg2  m  4
 
 X 1 
 X 2 


The combined uncertainty uc is
uc  u A2 y  u B2y  0,57 kgm2
Commentaries
The expanded uncertainty u is
u  2  uc  1,1 kgm2
The final result is given by
__________
BMI
 21,4  1,1 kgm-2
Conclusions
Computing of uncertainties of A and B
type is described in this paper
Computation 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/