Expression for uncertainty intervals handling skewness when the relative standard uncertainty is independent of the measurand level

Abstract

Uncertainty intervals for many measurement results are typically reported as symmetric intervals around the measured value. However, at large standard uncertainties (> approx. 15 %–20 %), it is necessary to consider asymmetry of the uncertainty intervals. Here, an expression for calculating uncertainty intervals handling asymmetry when the relative standard uncertainty is independent of the measurand level is presented. The expression is based on implementation of a power transformation (\({x}^{B}\)) for transformation of measurement results in order to achieve results that have a symmetric and approximate normal distribution. Uncertainty intervals are then calculated in the transformed space and back-transformed to the original space. The transformation includes a parameter, B, that needs to be optimized, and this can be based on real results, modelling of results, or on judgement. Two important reference points are B equal to 1 that corresponds to an approximate normal distribution of the original measurement results, and B approaching 0 that corresponds to an approximate log-normal distribution of the original measurement results. Comparisons are made with uncertainty intervals calculated using other expressions where it is assumed that measurement results have a normal distribution or a log-normal distribution. Implementation of the approach is demonstrated with several examples from chemical analysis.