1. Uncertainties of measurement in EXCEL
History and basic terms
Example
Conclusions
References
RNDr. Ctibor Henzl, Ph.D.
VŠB – Technical University Ostrava

2. 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

3. This often highly cited proposition about statistics is ascribed on Benjamin Disraeli, other times Lord Palmerston is regarded as the author
Benjamin Disraeli( ) Lord Palmerston( )

4. In his beautiful book „Moderní statistika, “Dr
In his beautiful book „Moderní statistika, “Dr. Helmut Swoboda indicates two reasons of this abusive statement:
Insufficient knowledge of methods, aims and facilities of the statistics
Many people consider statistics as something which is actually pseudostatistics

5. 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

6. 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 see [6] which is a source for a lot of recommendations and norms, see References.

7. Estimation of uncertainties of „A“ type
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
m is number of values, n is number of measurements

8. The sample variance of is calculated
Experimental standard deviation of the mean is used as a standard uncertainty of „A“ type

9. Uncertainties of „B“ type originate from systematic errors
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.

10. 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
If deviation cannot be practically exceeded  = 3, if deviaton can be exceeded  = 2

11. Gaussian law of propagation of uncertainties
Let us assume that y is a function of values x1, x2, … , xm
Uncertainty of the every value x1, x2, … ,xm contributes to the resulting uncertainty
The mean of y is

12. 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.

13. 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

14. There are sample covariance of means
in the adjacent diagonal of the matrix

15. Off-diagonal elements are zero, if variables x1,…,xn are not correlated. The result is then more simple
Gaussian law of propagation of uncertainties must again be applied when calculating uncertainties of type „B“.

16. Combined uncertainty
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.

17. 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%.

18. 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 were used.

19. 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.

20. BMI and health risk are related (see Table).

21. Table of
results
Gaussian law

22. Commentaries
Uncertainties of „A“ type are computed using
formulas
The formulas are based on statistical analysis of series of observations

23. Commentaries
Uncertainties of „B“ type are computed by
zmax is estimated,  is determined from distribution of probability, see table

24. Estimation of „B“ type uncertainties
Commentaries
Estimation of „B“ type uncertainties
in our example

25. Commentaries
The partial derivatives f/xi are calculated from their analytical expression
We use the Gaussian law of propagation of uncertainties for calculating uncertainties of „A“ type

26. Commentaries
The Gaussian law of propagation of uncertainties is used for calculating uncertainties of „B“ type
The combined uncertainty uc is

27. Commentaries
The expanded uncertainty u is
The final result is given by

28. Conclusions
Computing of uncertainties of A and B type is described in this paper
Computation has been made according to documents WECC doc and EAL-4/02
A practical-oriented example has been solved
Spreadsheet EXCEL has been used for computations

29. 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 : "Guidelines for Expression of the Uncertainty in Calibrations
[5]
[6]
[7] /JCGM/bibliography_on_uncertainty.html/