I’m learning about inequality measures, there are several ways to calculate it and I understand all but one.

$\kappa = \frac{E|X-X^\prime|}{2E(X)}$

I’m not sure what the second x means in the case of random variables.

up vote 4 down vote accepted

The numerator of the Gini index is the expected value of the difference, in absolute value, between the earnings (if $x$ denotes earnings) of two random individuals in the population.

In mathematical terms, the expectation here is taken over two random variables $x$ and $x'$, taken as independent and identically distributed. If I rewrite the Gini with $x_1$ and $x_2$ for convenience, and if $F(.)$ denotes the cdf of these random variables: $$ G = \frac{1}{E(X)}\int \int |x_1 - x_2|\ dF(x_1)\ dF(x_2), $$ where: $$ E(X) = \int x \ dF(x). $$

You could find more information on the Wikipedia page of the Gini index.

  • 3
    See also, quite independently of income inequality and so forth, the ideas of $L$-moments. – Nick Cox Sep 24 at 13:23

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