Gini Coefficient from Quintile Shares (Lorenz Curve by Trapezoids)

The Gini coefficient measures how unevenly a total (income, wealth, land) is shared. Plot the cumulative share held by the poorest fraction of the population against that fraction and you have the Lorenz curve, which is the diagonal when everyone holds the same. The coefficient is a definition, $G = 1 - 2B$, where $B$ is the area under the Lorenz curve. Lorenz (1905), the source listed here, wrote on methods of measuring the concentration of wealth.

This page works from the usual published table: the cumulative share held by the poorest 20 %, 40 %, 60 % and 80 %. Those four points and the two ends of the line are joined by straight lines, the area $B$ is found by trapezoids, and the algebra collapses to $G = 0.8 - 0.4\,(L_{20} + L_{40} + L_{60} + L_{80})$. Enter cumulative shares as fractions, such as 0.05 for 5 %, not the share of each fifth taken alone.

The two extremes check the formula. Equal shares, 0.2, 0.4, 0.6 and 0.8, put the curve on the diagonal and give $G = 0$. If the top fifth holds everything, the four lower points are all zero and $G = 0.8$, the most that fifths can show. In between, shares of 5, 10, 15, 25 and 45 per cent accumulate to 0.05, 0.15, 0.30 and 0.55, the trapezoid area is 0.31, and $G = 1 - 2 \times 0.31 = 0.38$. Solved backward, a $G$ of 0.38 with the first three points needs $L_{80} = 0.55$.

The result is an approximation to the Gini coefficient, not the coefficient itself. Straight lines cut the corners of the true curve, so the figure sits slightly below the Gini an agency works from individual records, by about 0.02 to 0.05 for the spreads seen in real incomes. A lognormal spread with $\sigma = 0.9$, whose exact Gini is 0.4755, shows as 0.4384 from its fifths. The formula is the right tool for reading a table of quintile shares and for checking one; the mean-difference form works from the values themselves, with no loss from joining points by straight lines.