Uncertainties of measurement in EXCEL History in addition to basic terms Example References Conclusions
Kozlowski, Lori, Features Reporter has reference to this Academic Journal, PHwiki organized this Journal Uncertainties of measurement in EXCEL History in addition to basic terms RNDr. Ctibor Henzl, Ph.D. VB 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 as long as ms 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)
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In his beautiful book Moderní statistika, Dr. Helmut Swoboda indicates two reasons of this abusive statement: Insufficient knowledge of methods, aims in addition to facilities of the statistics Many people consider statistics as something which is actually pseudostatistics The International Committee as long as Weights in addition to Measures (CIPM) has committed itself since 70 years of the last century to create an unique methodology as long as processing in addition to evaluating results of measurement 1977 The International Committee as long as Weights in addition to Measures (CIPM) asked the International Bureau of Weights in addition to Measures (BIPM) to collaborate with national metrology institutes in addition to elaborate recommendations as long as the solution of the situation, i.e. recommendation of an uni as long as m approach 1980 recommendation is published as Recommendation INC-1 (1980), see . 1990 West European Calibration Committee issued document WECC Doc. 19-1990 see  which is a source as long as a lot of recommendations in addition to norms, see References.
Uncertainties of A type originate from r in addition to om 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 The sample variance of is calculated Experimental st in addition to ard deviation of the mean is used as a st in addition to ard uncertainty of A type 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 in as long as mation are Technical documentation Experience with relevant materials in addition to instruments Knowledge of previous data etc.
Guess zmax maximal deviation of value, appropriate to source z in addition to look up relevant probability distribution in the following table Estimation of uncertainties of B type Determine uncertainty of B type 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 The mean of y is Uncertainty of the every value x1, x2, ,xm contributes to the resulting uncertainty Uncertainty of , symbolized by uAy is calculated by means of Gaussian law of propagation of uncertainties. The Matrix as long as m of this law is very suitable as long as computer programming.
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 There are sample covariance of means in the adjacent diagonal of the matrix 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.
Combined uncertainty Combined uncertainty uCy includes both types of uncertainties. It is computed as the geometrical mean of uAy in addition to uBy Exp in addition to ed uncertainty Exp in addition to ed uncertainty u is the product of the combined uncertainty uCy in addition to the coverage factor k. Result The result can be written in the as long as m If k = 2, estimated values are approximately normally distributed in addition to exp in addition to ed 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%. Let us compute the Body mass index (BMI) of a group of students. Example The personal weighting-machine LUXA in addition to folding rule LOGAREX 38031 were used. A group of 15 students 15 years old was chosen among the first year students of the Telecommunication school in Ostrava
Body mass index is defined as the ratio of mass in kg in addition to the square of height in m. Mass m in addition to length l are the directly measured values. BMI is calculated as long as each student. The arithmetic mean, complete with uncertainty of measurement is calculated as long as the group. BMI in addition to health risk are related (see Table). Gaussian law Table of results
Uncertainties of A type are computed using as long as mulas Commentaries The as long as mulas are based on statistical analysis of series of observations Commentaries Uncertainties of B type are computed by zmax is estimated, is determined from distribution of probability, see table Commentaries Estimation of B type uncertainties in our example
The partial derivatives f/xi are calculated from their analytical expression Commentaries We use the Gaussian law of propagation of uncertainties as long as calculating uncertainties of A type The combined uncertainty uc is Commentaries The Gaussian law of propagation of uncertainties is used as long as calculating uncertainties of B type The exp in addition to ed uncertainty u is Commentaries The final result is given by
Computing of uncertainties of A in addition to B type is described in this paper Conclusions Computation has been made according to documents WECC doc. 19-1990 in addition to EAL-4/02 A practical-oriented example has been solved Spreadsheet EXCEL has been used as long as computations References  ISO, Guide to Expression of Uncertainty in Measurement (International Organisation as long as St in addition to ardization), Geneva, Switzerl in addition to , 1993  NIST Technical Note 1297, Guidelines as long as Evaluating in addition to Expressing the Uncertainty of NIST Measurement Results, US Government Printing Office, Washington, 1994 Publication Reference EAL-4/02 (European cooperation as long as Accreditation of Laboratories),1999  WECC Doc. 19-1990 : “Guidelines as long as Expression of the Uncertainty in Calibrations  http://physics.nist.gov/cuu/index.html  http://www.european-accreditation.org  http://www.bipm.org/CC/documents /JCGM/bibliography-on-uncertainty.html/
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