The Gini Coefficient

The Gini coefficient is a ratio of the areas on a Lorenz curve.

Figure 2: Gini Coefficient Example

It is also a measure of the inequality of a distribution. If the area between the line of perfect equality and Lorenz curve is A, and the area under the Lorenz curve is B, then the Gini coefficient is $A/(A+B)$. Since $A+B = 0.5$, the Gini coefficient, $G = A/(.5) = 2A = 1-2B$. If the Lorenz curve is represented by the function $Y = L(X)$, the value of B can be found with integration and:

$G = 1 - 2\,\int_0^1 L(X) dX$

Some important properties of the Gini coefficient are:

1. The Gini coefficient is a measure of inequality of a distribution. It is defined as a ratio with values between 0 and 1: the numerator is the area between the Lorenz curve of the distribution and the uniform (perfect) distribution line; the denominator is the area under the uniform distribution line.
2. It is often used as a metric of inequality.
3. The higher the Gini coefficient, the greater the inequality.
4. A value of zero corresponds to perfect income equality (everyone has the same income), while a value of 1 corresponds to perfect income inequality (one person has all the income, and the rest of the population has none).
5. It is not affected by the shape of the Lorenz curve, only by the ratio of the areas used to compute it.
6. It does not indicate how the inequality is distributed, only the total amount of inequality.
7. The Gini coefficient can be used to indicate how a distribution changes over time and if this change shows that equality is increasing or decreasing.