1 / 29

Uncertainties of measurement in EXCEL

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

adanna
Download Presentation

Uncertainties of measurement in EXCEL

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 statisticalapproach 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(1804-1881) Lord Palmerston(1784-1865)

  4. 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 somethingwhich isactuallypseudostatistics

  5. The International Committee for Weights and Measures (CIPM) hascommitted 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 • 1980recommendation is published as Recommendation INC-1 (1980), see [5]. • 1990West European Calibration Committee issued document WECC Doc. 19-1990 see [6] which is a source for a lot of recommendations and norms, see References.

  7. 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“ typeoriginate 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, … ,xmcontributes 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 uis the product of the combined uncertainty uCyand 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 38031 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. 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 ofuncertainties for calculating uncertainties of „A“ type

  26. Commentaries • The Gaussian law of propagation ofuncertainties is usedfor 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 todocuments WECC doc. 19-1990 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. 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/

More Related