A normed measures of variability among proportions

Abstract

A measure of variability among a set of proportions is developed. There are no distributional assumptions, so the measure, H, is applicable in a wide variety of situations. H is scaled so that its range is zero (all proportions are equal) to one (maximum variability among the proportions, given the weighted average). The measure can be interpreted as a distance: the value of H indicates the position of the proportions relative to the possible extremes of no variability or maximum variability. Any set of constant weights can be applied to the proportions: the weights are used to compute the weighted average proportion and are also used to determine the extent to which each proportion affects the variability measure. Comparisons are made to other measures of variability and a numerical example is given.