On a curious bias arising when
the $\sqrt {\chi ^2/\nu }$ scaling prescription is first applied
to a sub-sample of the individual results

Giulio D'Agostini

Università “La Sapienza” and INFN, Roma, Italia

(giulio.dagostini@roma1.infn.it, http://www.roma1.infn.it/~dagos)

Abstract:

As it is well known, the standard deviation of a weighted average depends only on the individual standard deviations, but not on the dispersion of the values around the mean. This property leads sometimes to the embarrassing situation in which the combined result `looks' somehow at odds with the individual ones. A practical way to cure the problem is to enlarge the resulting standard deviation by the $\sqrt {\chi ^2/\nu }$ scaling, a prescription employed with arbitrary criteria on when to apply it and which individual results to use in the combination. But the `apparent' discrepancy between the combined result and the individual ones often remains. Moreover this rule does not affect the resulting `best value', even if the pattern of the individual results is highly skewed. In addition to these reasons of dissatisfaction, shared by many practitioners, the method causes another issue, recently noted on the published measurements of the charged kaon mass. It happens in fact that, if the prescription is applied twice, i.e. first to a sub-sample of the individual results and subsequently to the entire sample, then a bias on the result of the overall combination is introduced. The reason is that the prescription does not guaranty statistical sufficiency, whose importance is reminded in this script, written with a didactic spirit, with some historical notes and with a language to which most physicists are accustomed. The conclusion contains general remarks on the effective presentation of the experimental findings and a pertinent puzzle is proposed in the Appendix.

“Observations, for example, such as are distant from each other by an interval of a few
days $[$...$]$ are not to be used in the calculation as so many different positions, but it would
be better to derive from them a single place, which would be, as it were, a mean among all,
admitting, therefore, much greater accuracy than single observations considered separately.”
(F.C. Gauss, transl. by C.H. Davies)