Measuring social inequality with quantitative methodology: analytical estimates and empirical data analysis by Gini and $k$ indices

TL;DR

This paper develops analytical methods to quantify social inequality using Gini and $k$ indices, fitting empirical data with piecewise functions, and explores the relationships among different inequality measures.

Contribution

It introduces general formulas for inequality measures from known distributions and analyzes their bounds and relations, enhancing understanding of social inequality quantification.

Findings

Analytical formulas for Gini and $k$ indices derived from distribution functions.

Empirical data often fit better with piecewise functions with a crossover point.

Bounds of Gini index expressed as functions of the $k$ index.

Abstract

Social inequality manifested across different strata of human existence can be quantified in several ways. Here we compute non-entropic measures of inequality such as Lorenz curve, Gini index and the recently introduced $k$ index analytically from known distribution functions. We characterize the distribution functions of different quantities such as votes, journal citations, city size, etc. with suitable fits, compute their inequality measures and compare with the analytical results. A single analytic function is often not sufficient to fit the entire range of the probability distribution of the empirical data, and fit better to two distinct functions with a single crossover point. Here we provide general formulas to calculate these inequality measures for the above cases. We attempt to specify the crossover point by minimizing the gap between empirical and analytical evaluations of measures. Regarding the $k$ index as an `extra dimension', both the lower and upper bounds of the Gini index are obtained as a function of the $k$ index. This type of inequality relations among inequality indices might help us to check the validity of empirical and analytical evaluations of those indices.