On the poisson index of dispersion

A commonly used statistic for testing hypotheses concerning count data is the index of dispersion or variance test (c.f., Fisher 1950, Rao and Chakravarti 1956). The statistic is essentially the ratio of the sample variance to the sample mean,The hypothesis being tested is commonly referred to as the homogeneity hypothesis, often without further detailed specification. However, a closer study of the literature indicates that there are really two different hypotheses of interest.