Dispersion, Asymmetry and the Gini Index of Inequality

Abstract

studies have focused on the characteristics of one of the income inequality indices, the Gini Index. The purpose of this study is twofold: to present some additional, hitherto neglected, features of this Gini Index, and to show that some interesting new indices of location and skewness could be derived from its definition. The paper starts by presenting an illuminating algorithm which allows a quick computation of the Gini Index. On the basis of this algorithm, it is then proved that the Gini Index belongs to the family of relative mean deviations. The proof of this proposition is very much dependent on the construction of coefficients which have been labeled ci and shown to be central in the definition not only of Gini's well known mean-difference but also of some new indices like the G-mean. The G-mean is an index of location combining the features of the mean and the median, and another new index is an asymmetry index called AR, having properties which might make it more attractive than the coefficients of skewness generally used. Using the definition and properties of these various indices, we prove, in a final section, a certain number of interesting propositions concerning the Gini Index of Inequality. More precisely, it is shown that the Gini Index of symmetric distributions is always smaller than or equal to one half and that a necessary condition for the Gini Index to be greater than one half is that the distribution be skewed to the right.