Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/267030 
Year of Publication: 
2022
Series/Report no.: 
LIS Working Paper Series No. 830
Publisher: 
Luxembourg Income Study (LIS), Luxembourg
Abstract: 
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.
Subjects: 
Balance of Inequality
Balance of Inequality index
Barycenter
BOI index
Concentration
Concentration area
Concentration ratio
Gini index
Gini mean difference
Inequality
Income inequality
Lorenz curve
Pen parade
Quantile function
JEL: 
C10
C18
D31
D63
Document Type: 
Working Paper

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