Two simple measures of variability for categorical data

This paper proposes two new variability measures for categorical data. The first variability measure is obtained as one minus the square root of the sum of the squares of the relative frequencies of the different categories. The second measure is obtained by standardizing the first measure. The measures proposed are functions of the variability measure proposed by Gini [Variabilitá e Mutuabilitá Contributo allo Studio delle Distribuzioni e delle Relazioni Statistiche, C. Cuppini, Bologna, 1912] and approximate the coefficient of nominal variation introduced by Kvålseth [Coefficients of variation for nominal and ordinal categorical data, Percept. Motor Skills 80 (1995), pp. 843–847] when the number of categories increases. Different mathematical properties of the proposed variability measures are studied and analyzed. Several examples illustrate how the variability measures can be interpreted and used in practice.