Graphical representation of the Gini coefficient
The graph shows that the Gini coefficient is equal to the area marked A
divided by the sum of the areas marked A
, that is, Gini = A / (A + B)
. It is also equal to 2A
and to 1 − 2B
due to the fact that A + B = 0.5
(since the axes scale from 0 to 1).
The Gini coefficient is a single number aimed at measuring how far a country’s wealth distribution deviates from totally equal distribution of productivity.
The Gini coefficient is usually defined mathematically based on the Lorenz curve, which plots the proportion of the total income of the population (y axis) that is cumulatively earned by the bottom x% of the population (see diagram). The line at 45 degrees thus represents perfect equality of incomes. The Gini coefficient can then be thought of as the ratio of the area that lies between the line of equality and the Lorenz curve (marked A in the diagram) over the total area under the line of equality (marked A and B in the diagram); i.e., G = A / (A + B). It is also equal to 2A and to 1 − 2B due to the fact that A + B = 0.5 (since the axes scale from 0 to 1).
If all people have non-negative income (or wealth, as the case may be), the Gini coefficient can theoretically range from 0 (complete equality) to 1 (complete inequality); it is sometimes expressed as a percentage ranging between 0 and 100. In practice, both extreme values are not quite reached. If negative values are possible (such as the negative wealth of people with debts), then the Gini coefficient could theoretically be more than 1. Normally the mean (or total) is assumed positive, which rules out a Gini coefficient less than zero.
An alternative approach is to define the Gini coefficient as half of the relative mean absolute difference, which is mathematically equivalent to the Lorenz curve definition.
The mean absolute difference is the average absolute difference of all pairs of items of the population, and the relative mean absolute difference is the mean absolute difference divided by the average, to normalize for scale. if xi is the wealth or income of person i, and there are n persons, then the Gini coefficient G is given by:
When the income (or wealth) distribution is given as a continuous probability distribution function p(x), where p(x)dx is the fraction of the population with income x to x+dx, then the Gini coefficient is again half of the relative mean absolute difference:
where μ is the mean of the distribution and the lower limits of integration may be replaced by zero when all incomes are positive.