The Gmd: A Superior Measure of Variability for Non-Normal Distributions

Abstract

Of all measures of variability, the variance is by far the most popular. This paper argues that Gini's Mean Difference (GMD), an alternative index of variability, shares many properties with the variance, but can be more informative about the properties of distributions that depart from normality. Its superiority over the variance is important whenever one is interested in one or more of the following properties: (a) stochastic dominance: the GMD can be used to form necessary conditions for stochastic dominance, while the variance cannot; (b) exchangeability: the GMD has two correlation coefficients associated with it, the difference between them being sensitive to the exchangeability between the marginal distributions. This property may play an important role whenever the "index number" problem is severe, (i.e., whenever the choice of the base for comparison between two marginal distributions may determine the direction of the results), or whenever the investigation procedure is based on an optimization procedure; (c) stratification: when the overall distribution is composed of sub-populations, the GMD is sensitive to stratification among sub-populations. The paper surveys the properties of the two indices of variability and discusses their relevance to several fields of research.

Keywords: Variance, Correlation, ANOVA, Gini's MEAN Difference

JEL Classification: C10, C14