Measuring social inequality with quantitative methodology: Analytical estimates and empirical data analysis by Gini and kk indices

• Different social inequalities are measured using Lorenz curve, Gini index and kk index analytically.
• Use of suitable fits for characterization, inequality measures and compare with the analytical results.
• We provide general formulas to calculate inequality measures.
• Special technique is used to specify the crossover point between distributions.

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 kk 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 kk index as an ‘extra dimension’, both the lower and upper bounds of the Gini index are obtained as a function of the kk index. This type of inequality relations among inequality indices might help us to check the validity of empirical and analytical evaluations of those indices.