## Quick Reference

A graphical representation of social inequality introduced by the American economist Max Otto Lorenz (1876–1959) in 1905 in connection with the distribution of wealth in a population. Let *w* be the total income of those members of a population whose income is at most *v*, and let *W* be the total income of the population. In the Lorenz curve, *w/W* is plotted against the (cumulative) proportion of the population that has income at most *v*. So, if f is the probability density function of income then the curve is the plot of . The ratio of the area between the Lorenz curve and the 45° line to the total area below that line is the Gini index (also called the Gini coefficient), *G*.

Suppose a sample of *n* individuals have incomes *x*_{(1)}≤*x*_{(2)}≤…≤*x*_{(n)}. The Lorenz curve is approximated by the polygon joining the origin to the successive points with coordinates . If the corresponding income proportions are *y*_{(1)}≤*y*_{(2)}≤…≤*y*_{(n)} then *G* can be calculated using

**Lorenz curve**. The curves shown result from Pareto distributions of income with parameter *k* equal to one and parameter *a* taking the values 1, 1.2, 1.6, and 2.

**From:**
Lorenz curve
in
A Dictionary of Statistics »

*Subjects:*
Social Sciences — Probability and Statistics.

## Related content in Oxford Index

##### Reference entries

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.