To Örebro University

The Birge ratio is applied in metrology to enlarge quoted uncertainties when combining inconsistent measurement results on the same measurand. We discuss the statistical model underlying such a procedure and argue that the resulting uncertainty associated with the adjusted value is underrated. We provide a simple modification of this uncertainty on the basis of an objective Bayesian inference. While the proposed uncertainty approaches that obtained by the conventional procedure for a large number n of combined measurement results, differences are significant for small n. For example, for n = 4 we get an increase of 73% in the standard uncertainty associated with the adjusted value, and for n = 10 the increase is still 13%. We derive the posterior distribution for the adjusted value in closed form, including a 95% credible interval. In addition, we show that our results do not only hold when the distribution of the measurement results is assumed to be Gaussian, but for a whole family of (elliptically contoured) location-scale distributions. We illustrate the modified Birge method by its application to data from the 2002 adjustment of the Newtonian constant of gravitation.