Lorenz Curve — Definition, Formula & Examples

A Lorenz Curve is a graph that shows how evenly a quantity (such as income or wealth) is distributed across a population by plotting the cumulative share of the population against the cumulative share of that quantity.

The Lorenz Curve is a continuous, non-decreasing, convex function $L(p)$ defined on $[0, 1]$ , where $p$ represents the cumulative proportion of the population (ordered from lowest to highest value) and $L(p)$ represents the cumulative proportion of the total quantity held by that bottom $p$ fraction. Perfect equality corresponds to $L(p) = p$ , and the degree of bowing below this diagonal measures inequality.

How It Works

To construct a Lorenz Curve, sort all individuals from lowest to highest value (e.g., income). Then calculate cumulative percentages: for each fraction of the population, find what fraction of total income they collectively earn. Plot these pairs on a graph where the x-axis is cumulative population share and the y-axis is cumulative income share. A 45° diagonal line represents perfect equality — everyone earns the same. The further the Lorenz Curve bows below this diagonal, the greater the inequality. The area between the diagonal and the Lorenz Curve, relative to the total area under the diagonal, gives the Gini coefficient.

Worked Example

Problem: Five households earn $10,000, $20,000, $30,000, $40,000, and $100,000 respectively. Plot the key points of the Lorenz Curve.

Step 1: Find the total income and sort households from lowest to highest (already sorted).

\text{Total} = 10{,}000 + 20{,}000 + 30{,}000 + 40{,}000 + 100{,}000 = 200{,}000

Step 2: Compute cumulative population share and cumulative income share for each household.

\begin{array}{ccc} \text{Pop. %} & \text{Cumul. Income} & \text{Income %} \\ 20\% & 10{,}000 & 5\% \\ 40\% & 30{,}000 & 15\% \\ 60\% & 60{,}000 & 30\% \\ 80\% & 100{,}000 & 50\% \\ 100\% & 200{,}000 & 100\% \end{array}

Step 3: Plot the points (0, 0), (0.2, 0.05), (0.4, 0.15), (0.6, 0.30), (0.8, 0.50), (1.0, 1.0) and connect them. Compare to the 45° line of perfect equality.

Answer: The Lorenz Curve bows well below the equality line. For instance, the bottom 80% of households earn only 50% of total income, revealing significant inequality.

Visualization

Why It Matters

Economists and policy analysts use Lorenz Curves to visualize income and wealth inequality within countries. In AP Statistics and introductory economics courses, understanding this curve is essential for interpreting the Gini coefficient and evaluating the effects of taxation or redistribution policies.

Common Mistakes

Mistake: Forgetting to sort the data from lowest to highest before computing cumulative shares.

Correction: The Lorenz Curve requires data ordered from the smallest to the largest value. Without sorting, the cumulative percentages will be meaningless and the curve will not have its characteristic convex shape.