Calculation of key comparison reference values in the presence of non-zero-mean uncertainty distributions, using the maximum-likelihood technique

The international metrological community is involved in an ongoing series of comparisons of measurement standards. The calculation of differences from a “reference value”, some form of weighted mean of the results, has been recommended as a convenient figure of merit for participating laboratories. In temperature metrology, the uncertainties of the defining reference points are dominated by the estimate of the purity of the metal sample used for the fixed-point determinations. As impurities result mainly in a depression of the apparent phase-transition temperature, the corresponding uncertainty has a non-zero mean, and the simple “least-squares” type of statistical technique for combination of uncertainties is inadequate. A robust solution based on the maximum-likelihood technique is proposed. The technique is applied to published data from a triple point of water cell comparison, and it is shown that when an exponential distribution is chosen, the calculated estimate is relatively insensitive to the magnitude of the estimated impurity uncertainty.