The variance of a consensus

Abstract

In proficiency tests the consensus of the participants' results is often used as the assigned value to calculate z-scores. Where the consensus is quantified as the robust mean $\hat\mu_{{\rm rob}}$ of n results, the standard error of the assigned value is often taken to be ${{\hat\sigma_{{\rm rob}}}\mathord{\left/{\vphantom{{\hat\sigma_{{\rm rob}}}{\sqrt n}}}\right. \kern-\nulldelimiterspace}{\sqrt n}}$, where $\hat\sigma_{{\rm rob}}$ is the robust standard deviation estimated from the same data ¹ . As some of the results are downweighted in robust estimation, $\sqrt n$ is too large a denominator, so that ${{\hat\sigma_{{\rm rob}}}\mathord{\left/{\vphantom{{\hat\sigma_{{\rm rob}}}{\sqrt n}}}\right. \kern-\nulldelimiterspace}{\sqrt n}}$ tends to have a somewhat low bias. This bias is shown to be inconsequential for proficiency testing purposes. However, an unbiased estimate can be obtained by using the bootstrap.