Similarity, dissimilarity and exceptionality: generalizing Gini’s transvariation to measure “differentness” in many distributions

Following the work of Gini, Dagum and Tukey, this paper extends Gini’s Transvariation measure for comparing two distributions to the simultaneous comparison of many distributions. In so doing, it develops measures of absolute and relative similarity, dissimilarity and exceptionality together with techniques for assessing particular aspects of variations across those distributions. These techniques are exemplified in a study of differences between the income distributions of males and females drawn from Metis, Inuit, North American Indian and Non-Aboriginal constituencies in Canada in the first decade of the twenty-first century. While the distributions were becoming increasingly similar (interpreted as improving equality of opportunity), this was occurring primarily at the center of the distribution. At the extremes, the distributions were diverging, suggesting that such improvements in equality of opportunity were not for all.