Several statistical tests are available for testing the Poisson hypothesis and/or the equidispersion of a point process. The capability to discriminate between the Poissonian behaviour and more complex processes is fundamental in many areas of research including earthquake analysis, hydrology, ecology, biology, signal analysis and sociology. This study investigates the relationship between two ... [Show full abstract] indices often used for detecting departures from equidispersion, namely, the index of dispersion (ID) and the Allan factor (AF). Since an approximation of the sampling distribution of AF for Poisson data has been recently proposed in the literature, we perform a detailed analysis of its properties and its relationship with the asymptotic sampling distribution of ID. Moreover, the statistical power of the AF for testing the Poisson hypothesis is assessed by using an extensive Monte Carlo simulation, and the performances of AF and ID are compared. We propose a simplified version of the AF sampling distribution that does not depend on the rate of occurrence keeping the maximum errors of extreme percentiles always smaller than 2–3 %. The power study highlights that ID systematically outperforms AF for discriminating between equidispersed and under/over dispersed data. Both indices show the same lack of power for distinguishing between data drawn from equidispersed non-Poissonian distribution functions. Therefore, even though AF is a useful statistical tool for detecting the possible fractal behaviour of point processes, ID should be preferred when the analysis aims at assessing equidispersion. The lack of power for a small sample size confirms the difficulty of identifying the true nature of the occurrence process of rare events.