The barycenter of the distribution and its application to the measurement of inequality: The balance of inequality, the Gini index, and the Lorenz curve

This paper introduces in statistics the notion of the barycenter of the distribution of a non-negative random variable Y with a positive finite mean μY and the quantile function Q(x). The barycenter is denoted by μX and defined as the expected value of the random variable X having the probability density function fX(x) = Q(x)/μY. For continuous populations, the Gini index is 2μX - 1, i.e., the normalization of the barycenter, which is in the range [0, 1/2], the concentration area is μX - 1/2, and the Gini's mean difference is 4μY (μX - 1/2). The same barycenter-based formulae hold for normalized discrete populations. The introduction of the barycenter allows for new economic, geometrical, physical, and statistical interpretations of these measures. For income distributions, the barycenter represents the expected recipient of one unit of income, as if the stochastic process that leads to the distribution of the total income among the population was observable as it unfolds. The barycenter splits the population into two groups, which can be considered as 'the winners' and 'the losers' in the income distribution, or 'the rich' and 'the poor'. We provide examples of application to thirty theoretical distributions and an empirical application with the estimation of personal income inequality in Luxembourg Income Study Database's countries. We conclude that the barycenter is a new measure of the location or central tendency of distributions, which may have wide applications in both economics and statistics.