Lorenz curve

Sometimes the entire Lorenz curve is not known, and only values at certain intervals are given.

The Lorenz curve cannot rise above the line of perfect equality.

The distribution of income within a community may be represented by the Lorenz curve.

This result can be derived from the Lorenz curve formula given below.

The Lorenz curve for a probability distribution is a continuous function.

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

If the variable being measured cannot take negative values, the Lorenz curve:

Points on the Lorenz curve represent statements like "the bottom 20% of all households have 10% of the total income."

The Lorenz curve is not defined if the mean of the probability distribution is zero or infinite.

The Lorenz curve is invariant under positive scaling.

Within a society can be measured by various methods, including the Lorenz curve and the Gini coefficient.

The Gini index is defined as a ratio of the areas on the Lorenz curve diagram.

The Lorenz curve is closely associated with measures of income inequality, such as the Gini coefficient.

Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line.

A better known inequality measure is the Gini coefficient which is also based on the Lorenz curve.

Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.

If the Lorenz curve is differentiable:

In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the Lorenz curve.

The relative mean difference is equal to twice the Gini coefficient which is defined in terms of the Lorenz curve.

If the Lorenz curve is twice differentiable, then the probability density function f(x) exists at that point and:

The most commonly used measures are quantile shares and Lorenz curves (and associated Gini coefficients).

These will typically be presented graphically as a Lorenz curve or in the form of an index such as the Gini coefficient.

In that case, the Gini coefficient can be approximated by using various techniques for interpolating the missing values of the Lorenz curve.

The term Lorenz curve seems first to have been used in 1912 in a textbook The Elements of Statistical Method.

The information in a Lorenz curve may be summarized by the Gini coefficient and the Lorenz asymmetry coefficient.