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# Lorenz curve

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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.

Subjects: Social Sciences — Probability and Statistics.

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