# Gini coefficient

## What is the Gini coefficient?

Corrado Gini invented a solution to handle cases in which the Lorenz criterion is unable to rank societies: the Gini coefficient. In this way, the Gini coefficient is a completion of the Lorenz ordering. Furthermore, when Lorenz curves succeed in ranking societies, the Gini coefficient produces the same rankings, making the Gini coefficient Lorenz consistent.

To calculate the Gini coefficient, let \(A\) denote the area between the Lorenz curve and the line of perfect equality. Let \(B\) denote the area below the Lorenz curve. The Gini coefficient is the ratio of \(A\) to \(A + B\):

\[\text{Gini} = \frac{A}{A + B}\]The Gini coefficient ranges from 0 to 1. A Gini coefficient of 0 corresponds to perfect equality and a Gini coefficient of 1 corresponds to the highest possible level of inequality.

In order to use this ratio, we need to compute the areas denoted by \(A\) and \(B\). In the figures below, we demonstrate how you can split up the \(B\) area into triangles and trapezoids to make the computation easier.

We can use a simple formula to find \(B\) from \(A\), or \(A\) from \(B\):

\[A + B = 0.50 \rightarrow A = 0.50 − B\]and

\[B = 0.50 − A\]This means that we only need to find the area of \(A\) or \(B\) using our Lorenz curve.

You can use use the following tool to calculate the Gini coefficient using the area method:

Enter the income distribution, using commas to separate the income values.

- The \(B\) area is
- The \(A\) area is \(0.50 - B =\)

equation

## Formula method

Using areas to compute the Gini coefficient can be cumbersome when there are many points in the Lorenz curve. In these cases, we use a mathematical formula to calculate the Gini coefficient. The Gini coefficient is exactly equivalent to:

\[G = \frac{1}{2 \mu n^{2}} \sum^{n}_{i = 1} \sum^{n}_{j = 1} |y_{i} − y_{j}|\]where the incomes are \(y_{1}, \ldots ,y_{n}\) with mean income \(\mu\) and population size \(n\).

You can use the following tool to calculate the Gini coefficient using the formula method:

Enter the income distribution, using commas to separate the income values.

equation