Measuring the Dispersion of Discrete Distributions

Measuring dispersion is among the most fundamental and ubiquitous concepts in statistics, both in applied and theoretical contexts. In order to ensure that dispersion measures like the standard deviation indeed capture the dispersion of any given distribution, they are by definition required to preserve a stochastic order of dispersion. The most basic order that functions as a foundation underneath the concept of dispersion measures is the so-called dispersive order. However, that order is incompatible with almost all discrete distributions, including all lattice distributions and most empirical distributions. Thus, there is no guarantee that popular measures properly capture the dispersion of these distributions. In this paper, discrete adaptations of the dispersive order are derived and analyzed. Their derivation is directly informed by key properties of the dispersive order in order to obtain a foundation for the measurement of discrete dispersion that is as similar as possible to the continuous setting. Two slightly different orders are obtained that both have numerous properties that the original dispersive order also has. Their behaviour on well-known families of lattice distribution is generally as expected if the parameter differences are large enough. Most popular dispersion measures preserve both discrete dispersive orders, which rigorously ensures that they are also meaningful in discrete settings. However, the interquantile range preserves neither discrete order, yielding that it should not be used to measure the dispersion of discrete distributions.