  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.
