A Rapid Test for the Poisson Distribution Using the Range

A rapid approximate test based on the range is presented as an alternative to the usual index of dispersion method for examining small samples for conformity to the Poisson distribution. Let X1, ⋯, Xkbe a sample of size k from a Poisson distribution. Then it is shown using a result of Johnson and Young that the approximate percentiles Rpof the conditional distribution of R = max Xj- min Xj, given the sample mean X̄, are given by R$_p = \sqrt{\bar{X}}w_p$, where the wpare the percentiles of the distribution of w, the range of k independent unit normal variates. Graphical methods are presented which not only facilitate the application of the test but provide valuable insight into possible causes for deviation from the Poisson distribution. The procedures are applied to several examples.