Gini Coefficient from the Mean Absolute Difference

The Gini coefficient is one number for how unevenly something is shared out: 0 when everyone has the same, close to 1 when one holder has everything. This page defines it through the mean absolute difference. Pick two members at random and take how far apart their amounts are; averaging that over every possible pair gives $\Delta$, and dividing by twice the mean $\mu$ gives $G$. It is a definition, not a law or a fit, and it is the same coefficient that is defined from the Lorenz curve. Gini introduced the mean difference in 1912, in Variabilità e mutabilità.

The double sum in $\Delta$ runs over every ordered pair of members, $n^2$ terms in all, so each pair is counted both ways and each member is paired with itself. That is why the sum has 36 terms for six people. Because $\Delta$ and $\mu$ are in the same unit (dollars, hectares, kWh), the ratio has no unit. Reach for this form when you have the individual values; when you have only a table of shares by fifths of the population, the quintile-share form of the index is the one to use. It works for income, wealth, land, energy use or the catch per boat, and 0.3 to 0.5 is typical for incomes.

Take the six-person population 1, 2, 3, 4, 5, 15. The mean is 5, and the sum of $|x_i - x_j|$ over all 36 ordered pairs is 160, so $\Delta = 160/36 = 4.444444$ and $G = 4.444444/(2 \times 5) = 0.444444$. Two other routes agree: the sorted-values form and the Lorenz trapezoids both give 0.444444. The relation also rearranges. With $G = 0.45$ and a mean of 32 000, $\Delta = 2\mu G = 28\,800$. With $\Delta = 18$ and $G = 0.36$, $\mu = \Delta/(2G) = 25$.

The condition to remember is that no amount may be negative. Like any single summary, the number says how unequal the sharing is but not where in the population the gap lies.