The Gini coefficient is a numerical statistic used to measure income inequality in a society. It was developed by Italian statistician and sociologist Corrado Gini in the early 1900s.

## The Lorenz Curve

In order to calculate the Gini coefficient, it's important to first understand the Lorenz curve, which is a graphical representation of income inequality in a society. A hypothetical Lorenz curve is shown in the above diagram.

## Calculating the Gini Coefficient

Once a Lorenz curve is constructed, calculating the Gini coefficient is pretty straightforward. The Gini coefficient is equal to A/(A+B), where A and B are as labeled in the diagram above. (Sometimes the Gini coefficient is represented as a percentage or an index, in which case it would be equal to (A/(A+B))x100%.)

As stated in the Lorenz curve article, the straight line in the diagram represents perfect equality in a society, and Lorenz curves that are further away from that diagonal line represent higher levels of inequality. Therefore, larger Gini coefficients represent higher levels of inequality and smaller Gini coefficients represent lower levels of inequality (i.e. higher levels of equality).

In order to mathematically calculate the areas of regions A and B, it is generally necessary to use calculus to calculate the areas below the Lorenz curve and between the Lorenz curve and the diagonal line.

## A Lower Bound on the Gini Coefficient

The Lorenz curve is a diagonal 45-degree line in societies that have perfect income equality. This is simply because, if everyone makes the same amount of money, the bottom 10 percent of people make 10 percent of the money, the bottom 27 percent of people make 27 percent of the money, and so on.

Therefore, the area labeled A in the previous diagram is equal to zero in perfectly equal societies. This implies that A/(A+B) is also equal to zero, so perfectly equal societies have Gini coefficients of zero.

## An Upper Bound on the Gini Coefficient

Maximum inequality in a society occurs when one person makes all of the money. In this situation, the Lorenz curve is at zero all the way out until the right-hand edge, where it makes a right angle and goes up to the top right corner. This shape occurs simply because, if one person has all of the money, society has zero percent of the income until that last guy is added in, at which point it has 100 percent of the income.

In this case, the region labeled B in the earlier diagram is equal to zero, and the Gini coefficient A/(A+B) is equal to 1 (or 100%).

## The Gini Coefficient

In general, societies experience neither perfect equality nor perfect inequality, so Gini coefficients are typically somewhere between 0 and 1, or between 0 and 100% if expressed as percentages.