Patterns and Approaches for Reexamining the Guide to the Expression of Uncertainty in Measurement. Part 1. Shortcomings
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GENERAL PROBLEMS OF METROLOGY AND MEASUREMENT TECHNIQUE PATTERNS AND APPROACHES FOR REEXAMINING THE GUIDE TO THE EXPRESSION OF UNCERTAINTY IN MEASUREMENT. PART 1. SHORTCOMINGS OF THE GUIDE AND SUPPLEMENT 1
E. A. Golubev
UDC 620.1.08
The shortcomings of the ISO Guide to the Expression of Uncertainty in Measurement that most often lead to erroneous results when using the error propagation law for estimating the uncertainty of measurements are systematized. A new interpretation of the concept of random measurement error is given. The consistency of measures of uncertainty and precision of measurements is confirmed analytically. Keywords: sets, random error, influence factors, subjective probabilities, nonlinearity of transfer function.
Difficulties arise when using the ISO Guide to the Expression of Uncertainty in Measurement (referred to below as the Guide) [1] because of the unsubstantiated rules stated there for evaluating uncertainty and their excessive complexity. These rules should be a consequence of a model for the appearance of uncertainty that corresponds to the assumptions given below. The Law of Uncertainty Propagation. According to the Guide, for indirect measurements the estimated dispersion 2 s (y) characterizing the uncertainty in the measurement of the output value y must be written in the form of a weighted sum of ordinary sample dispersions1 Δx 2k = ( xik − xik )2 and covariances Δx k Δxl = ( xik − xik )( xil − xil ) that correlate the output values xk (the law of uncertainty propagation): N
2
N −1
⎛ ∂y ⎞ 2 s ( y) ≈ ⎜ dx ⎟ Δx k + 2 ⎝ ⎠ k k =1 k =1 2
∑
N
∑ ∑
l = k +1
∂y ∂y Δx Δx . dx k dxl k l
(1)
Here y = ƒ(xk) is a function that reflects the dependence of the output value y on the input quantities in an indirect measurement (the transfer function) and the average is taken over sets of measured values of the output quantities xk. Thus, when esti-
1
The sample dispersions are obtained over a finite set and are based on the sample mean.
All-Russia Research Institute of Metrological Service (VNIIMS), Moscow, Russia; e-mail: [email protected]. Translated from Izmeritel’naya Tekhnika, No. 11, pp. 10–14, November, 2012. Original article submitted September 10, 2012.
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0543-1972/13/5511-1240 ©2013 Springer Science+Business Media New York
mating the uncertainty it is necessary in principle to determine a series of ordinary dispersions and even more covariances of the input quantities. In the Guide (and in metrology, as a whole), essentially no use is made of the concept of a set over which the average is taken in calculating the statistical characteristics of the random quantities dealt with by metrologists in measuring and processing data. Here we speak, not only of sets of measured values or observations, but also of sets of values of one or another statistical characteristic obtained in non-infinite samples.2 This is not only the set of influence factors (influencing the results of measuring and processing data), but also the set of manifestations for each of them and for their combi
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