On combining significances. Some trivial examples

TL;DR

The paper proposes simple formulas for calculating and combining statistical significances in Poisson experiments, including cases with systematic errors, to improve analysis of experimental results.

Contribution

It introduces straightforward methods for significance calculation and combination, including systematic errors, for Poisson-distributed data.

Findings

Significance formula $S = (n_{obs}-\lambda)/\sqrt{\lambda}$ for large Poisson parameters

Combination rule for significances $S_{comb}$ using weighted averages

Extension of significance combination to include systematic errors

Abstract

For Poisson distribution $Pois(n, \lambda)$ with $\lambda \gg 1$, $n \gg 1$ we propose to determine significance as $S = \frac{n_{obs}-\lambda}{\sqrt{\lambda}}$. The significance $S$ coincides up to sign with often used significance. For experiments which measure the same quantities the natural but not unique rule for significance combining is $S_{comb}(S_1, S_2) = \frac{S_1\sigma_1+S_2\sigma_2}{\sqrt{\sigma^2_1+\sigma^2_2}}$, where $\sigma_1$ and $\sigma_2$ are variations. We also propose the rule for significances combining for the case with systematic errors.