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.

翻译：离散度的度量是统计学中最基础且普遍存在的概念之一，无论是在应用还是理论背景下。为了确保标准差等离散度度量确实能够捕捉任何给定分布的离散程度，根据定义，它们必须保持离散度的随机序。作为离散度度量概念基础的最基本序是所谓的离散序。然而，该序与几乎所有离散分布都不兼容，包括所有格点分布和大多数经验分布。因此，无法保证常用度量能够恰当地捕捉这些分布的离散度。本文推导并分析了离散序的离散适应形式。其推导直接参考了离散序的关键性质，旨在为离散离散度的度量建立一个尽可能接近连续情形的理论基础。我们得到了两个略有不同的序，它们都具有原始离散序所具有的众多性质。在已知的格点分布族上，当参数差异足够大时，它们的行为通常符合预期。大多数常用的离散度度量都保持了这两种离散离散序，这严格确保了它们在离散背景下也是有意义的。然而，分位数间距既不保持任何一种离散序，这表明它不应用于度量离散分布的离散度。