Lorenz Curve - Calculation

Calculation

The Lorenz curve can often be represented by a function L(F), where F is represented by the horizontal axis, and L is represented by 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 F₀ = 0, L₀ = 0, and for i = 1 to n:

For a discrete probability function f(y), let y_i, i = 1 to n, be the points with non-zero probabilities indexed in increasing 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 F₀ = 0, L₀ = 0, and for i = 1 to n:

For a probability density function f(x) with the cumulative distribution function F(x), the Lorenz curve L(F(x)) is given by:

where denotes the average. For a cumulative distribution function F(x) with inverse x(F), the Lorenz curve L(F) is given by:

The inverse x(F) may not exist because the cumulative distribution function has intervals of constant values. However, the previous formula can still apply by generalizing the definition of x(F):

x(F₁) = inf {y : F(y) ≥ F₁}

For an example of a Lorenz curve, see Pareto distribution.