  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.
