Expediency of using “corrected” variances when evaluating measurement result uncertainty

Abstract

Background. When estimating the uncertainty of the measurement result, “corrected” variances are used. This leads to a violation of the rule for adding dispersions, which is of great practical importance in metrological calculations. The purpose of the work is a statistical test of the significance of the difference in sample variances obtained by averaging the sums of squared deviations both over the sample size and the number of degrees of freedom, compared with their difference from the general variance. The object of the study is small samples of a normally distributed random variable. The subject of the study is the average sums of squared deviations. Materials and methods. The studies were carried out using: the method of numerical simulation experiment (Monte Carlo method), the method of comparison (comparison), the method of testing statistical hypotheses. Results. Using a large number of samples (104) of a small volume of a standard normally distributed random variable, sample variances and standard deviations were determined with averaging over the sample size or over the number of degrees of freedom. They are compared with each other and with general parameters. Conclusions. In the case of small samples, typical for measurements by standardized methods, it was found: the presence of a bias of sample “corrected” standard deviation relative to the general one; statistical insignificance of the method of averaging the sums of squared deviations (by sample size or by degrees of freedom). The method of averaging over the sample size leads to the exact implementation of the rule of addition of variances and, therefore, is more preferable when calculating variances in the top-down method for estimating the uncertainty of measurement results.

Keywords

• variance
• uncertainty
• estimate bias
• simulation experiment
• statistical hypothesis