Extreme values, means, and inequality measurement

We examine some ordinal measures of inequality that are familiar from the literature. These measures have a quite simple structure in that their values are determined by combinations of specific summary statistics such as the extreme values and the arithmetic mean of a distribution. In spite of their common appearance, there seem to be no axiomatizations available so far, and this paper is intended to fill that gap. In particular, we consider the absolute and relative variants of the range; the max-mean and the mean-min orderings; and quantile-based measures. In addition, we provide some empirical observations that are intended to illustrate that, although these orderings are straightforward to define, some of them display a surprisingly high correlation with alternative (more complex) measures.