The Lorenz Curve

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

The Lorenz curve is compared with the perfect equality line, which is a linear relationships that plots a distribution where each element has an equal value in its shares of X and Y. For instance, in a distribution of 10 elements, if there is perfect equality, the 5th element would have a cumulative percentage of 50% for X and Y. The perfect equality line forms an angle of 45 degrees with a slope of 100/N. The perfect inequality line represents a distribution where one element 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).