The Lorenz Curve

The Lorenz Curve

The Lorenz curve is a graphical representation of the proportionality of a distribution; the cumulative percentage of the values. To construct a Lorenz curve, all the observations of a distribution must be ordered from the most important to the least important. Then, each observation is plotted according to their cumulative percentage of X and Y; X being the cumulative percentage of observations and Y being their cumulative importance. For instance, out of a distribution of 10 observations (N), the first observation would represent 10% of X and whatever percentage of Y it represents (this percentage must be the highest in the distribution). The second observation would cumulatively represent 20% of X (its 10% plus the 10% of the first observation) and its percentage of Y plus the percentage of Y of the first observation.

The Lorenz curve is compared with the perfect equality line, which is a linear relationship that plots a distribution where each observation has an equal value in its shares of X and Y. For instance, in a distribution of 10 observations, if there is perfect equality, the 5th observation would have a cumulative percentage of 50% for X and Y. The perfect inequality line represents a distribution where one observation has the total cumulative percentage of Y while the others have none.

The Gini coefficient is defined graphically as a ratio of two surfaces involving the summation of all vertical deviations between the Lorenz curve and the perfect equality line (A) divided by the difference between the perfect equality and perfect inequality lines (A+B).