Lorenz curve

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The Lorenz curve is a graphical representation of the cumulative distribution function of a probability distribution; it is a graph showing the proportion of the distribution assumed by the bottom y% of the values. It is often used to represent income distribution, where it shows for the bottom x% of households, what percentage y% of the total income they have. The percentage of households is plotted on the x-axis, the percentage of income on the y-axis. It can also be used to show distribution of assets. In such uses, some political doctrines (e.g. socialism) consider it to represent social inequality. It was developed by Max O. Lorenz in 1905 for representing income distribution.

Image:Lorenz-curve1.png

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Explanation

Every point on the Lorenz curve represents a statement like "the bottom 20% of all households have 10% of the total income". A perfectly equal income distribution would be one in which every person has the same income. In this case, the bottom N% of society would always have N% of the income. This can be depicted by the straight line y = x; called the line of perfect equality or the 45° line.

By contrast, a perfectly unequal distribution would be one in which one person has all the income and everyone else has none. In that case, the curve would be at y = 0 for all x < 100, and y = 100 when x = 100. This curve is called the line of perfect inequality.

If the variable being measured cannot take negative values, it is impossible for the Lorenz curve to rise above the line of perfect equality, or sink below the line of perfect inequality; it is increasing and convex to the y-axis.

The Gini coefficient is the area between the line of perfect equality and the observed Lorenz curve, as a percentage of the area between the line of perfect equality and the line of perfect inequality.

For any distribution, the Lorenz curve L(F)  is written in terms of the probability density function (f(x)) or the cumulative distribution function (F(x)) as:

<math>L(F)=\frac{\int_{-\infty}^{x(F)} xf(x)\,dx}{\int_{-\infty}^\infty xf(x)\,dx}

=\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'}</math>

where x(F) is the inverse of the cumulative distribution function F(x) (for example, see the Pareto distribution).

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References

Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association. 9: 209-219.

[also Will Dawson's contributions]

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See also

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External links

fr:Courbe de Lorenz pl:Krzywa Lorenza zh:劳伦茨曲线

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