On the Uncertainty Evaluation for Repeated Measurements

Abstract

Uncertainty evaluation for repeated measurements has been discussed previously by various researchers and two ways of proceeding were proposed, dependent upon whether or not there is a significative difference in the individual results of repeated measurements. In this paper, the analysis of variance perspective is used to clarify and unify the ideas of previous works. It is shown that the two above-mentioned approaches involve the use of two different estimators for the evaluation of the random component of the standard uncertainty. Based on this, the efficiency of the two estimators is compared in order to propose a procedure for correct and efficient use of previous approaches to perform the uncertainty evaluation for repeated measurements.

Annex: MATLAB Programme for Evaluating the Relative Standard Deviation of \({\user2{s(}}{{\bar{\user2{X}}}}_{\user2{A}}\user2{)}\) and \({\user2{s}^\prime}{\user2{(}{\bar{\user2{X}}}}_{\user2{A}}\user2{)}\)

The Monte Carlo Method (MCM) allows the evaluation of the probability distributions of the estimators $s\left( {\bar{X}_{A} } \right)$ and $s^{\prime}\left( {\bar{X}_{A} } \right)$, and consequently their relative standard deviations, based on the random sampling of $X_{A}$ and the evaluation of the estimators with the values found. The programme (Fig. 2) functions repeatedly sampling the probability distribution of the input quantity in sets of s samples (normally $s = 10000$). Each sample consists of m groups, each of n observations, for which the estimator of $u\left( {\bar{x}_{A} } \right)$, $s\left( {\bar{X}_{A} } \right)$ or $s^{\prime}\left( {\bar{X}_{A} } \right)$, is evaluated. Every time a set of samples is taken, the standard deviation associated with the average of the estimate for the set is calculated, together with the numerical tolerance associated with the standard deviation of the overall standard deviation. When this standard deviation exceeds the numerical tolerance, the process is repeated, taking another set of samples. Once this stabilization criterion is met (GUM Supplement 1 [9] 7.9) and thus the process is completed, the programme displays the results on screen: the estimate, its standard deviation and its relative standard deviation as a percentage. It also generates a histogram representing the probability density function of the statistic.

Fig. 2

Programme in MATLAB for the evaluation of the relative standard deviation of the standard uncertainty $u\left( {\bar{x}_{A} } \right)$, estimated by $s\left( {\bar{X}_{A} } \right)$ or $s^{\prime}\left( {\bar{X}_{A} } \right)$, for an input variable $X_{A}$ normally distributed

Note that although $s^{2} \left( {\bar{X}_{A} } \right)$ and $s^{'2} \left( {\bar{X}_{A} } \right)$ are unbiased estimators of $\sigma_{{\bar{X}_{A} }}^{2}$, $s\left( {\bar{X}_{A} } \right)$ and $s'\left( {\bar{X}_{A} } \right)$ are not unbiased estimators of $\sigma_{{\bar{X}_{A} }}$ for a finite number of observations; nevertheless, this fact is not relevant to the calculation of measurement uncertainty [13]. Note also that the relative standard deviations of $s\left( {\bar{X}_{A} } \right)$ and $s^{\prime}\left( {\bar{X}_{A} } \right)$ are independent of the expectation and standard deviation of the normal distribution assumed for $X_{A}$.