Formulas used in computing the Lorenz Curve and Gini Coefficient

The Lorenz curve can be represented by a function $L(F)$, where $F$ is the horizontal axis, and $L$ is the vertical axis.

For a population of size n, with a sequence of values $y_{i}$ $i = 1$ to $n$ that are indexed in non-decreasing order $(y_{i} <= y_{i+1})$ the Lorenz curve is the continuous piecewise linear function connecting the points ($F_i , L_i$), i = 0 to n, where $F_0 = 0, L_0 = 0$, and for i = 1 to n:

$F_{i} = i/n$
$S_{i}= \Sigma_{j=1}^i\ y_{j}$
$L_{i} = S_i / S_n$

The amount of inequality in two societies, or in two scenarios can be compared based on their Lorenz curves. If the curve in one case is farther away from the line of perfect equality for every value along the horizontal axis, then that case is considered to have less equality than a case with a curve nearer to the equality line.

The line of perfect equality in the Lorenz curve is $L(F) = F$ and represents a uniform distribution, or equality. In cases where the curve does not lie on the line of perfect equality, every point on the curve represents a statement like: "the bottom 20% of households has 10% of the total income".

The Gini coefficient is defined as a ratio of the areas on the Lorenz curve diagram. It is 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:

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

The higher the Gini coefficient, the greater the inequality.