The Gini Coefficient and the Lorenz Curve: Measuring Inequality in a Single Number

How do you summarize the entire income distribution of 132 million households — every earner from the dishwasher to the hedge-fund founder — in one number? Economists do it constantly, and the number they reach for is the Gini coefficient. The United States scores about 0.49 on it. Denmark scores around 0.28. South Africa, near 0.63. Those three numbers tell you, at a glance, that America's income is shared far less evenly than Denmark's and far more evenly than South Africa's — without showing you a single dollar figure. Understanding how that compression works, and what it conceals, is the difference between using the statistic and being fooled by it.

Start with the picture: the Lorenz curve

Before the single number comes a graph. The Lorenz curve is built by lining up every household from poorest to richest, then plotting the cumulative share of total income against the cumulative share of households. The x-axis runs from 0 to 100 percent of households; the y-axis runs from 0 to 100 percent of income.

If income were perfectly equal — everyone earning the same — the poorest 20 percent of households would earn exactly 20 percent of income, the poorest 50 percent would earn 50 percent, and so on. That traces a straight 45-degree diagonal: the line of perfect equality. Real distributions never follow it. Because the poor earn less than their proportional share, the actual Lorenz curve sags below the diagonal, bowing toward the bottom-right corner. The more unequal the society, the deeper the sag.

The U.S. Census Bureau defines the relationship precisely: the Gini index, as the Census explains, "is based on the difference between the Lorenz curve (the observed cumulative income distribution) and the notion of a perfectly equal income distribution." The bigger that difference — the larger the gap between the diagonal and the sagging curve — the more unequal the distribution.

From curve to number: the Gini coefficient

The Gini coefficient turns that visual gap into a single value. Picture two areas on the graph: area A, the crescent between the line of perfect equality and the Lorenz curve, and area B, everything below the Lorenz curve. The Gini is defined as A ÷ (A + B) — the share of the total triangle that the inequality crescent occupies.

The scale is intuitive. As the Census puts it, the coefficient "ranges from 0, indicating perfect equality (where everyone receives an equal share), to 1, perfect inequality (where only one recipient or group of recipients receives all the income)." A Gini of 0 means the Lorenz curve is the diagonal — area A vanishes. A Gini of 1 means one household has everything and the curve hugs the bottom axis until the very last point — area A swallows the whole triangle. Every real economy lands somewhere between.

Compute one yourself

The geometry is approachable enough to do by hand, which is the best way to make the number stop feeling like a black box. Let us compute a rough Gini from the kind of quintile data the Census publishes. Take these five income shares for the five fifths of households, ordered poorest to richest — close to the actual U.S. pattern:

The cumulative column gives us the Lorenz curve at five points: (0,0), (20,3), (40,11), (60,25), (80,47), (100,100), all in percentages.

Now find area B under the Lorenz curve using the trapezoid method — slice the curve into five vertical strips, each 20 percentage points wide, and treat each as a trapezoid whose area is (width) × (average of its two heights). Working in proportions (so widths are 0.2 and heights are the cumulative income shares as decimals):

• Strip 1: 0.2 × (0 + 0.03)/2 = 0.003
• Strip 2: 0.2 × (0.03 + 0.11)/2 = 0.014
• Strip 3: 0.2 × (0.11 + 0.25)/2 = 0.036
• Strip 4: 0.2 × (0.25 + 0.47)/2 = 0.072
• Strip 5: 0.2 × (0.47 + 1.00)/2 = 0.147

Sum: B ≈ 0.272. The total area under the equality diagonal is 0.5 (it's a triangle with base 1 and height 1). So area A = 0.5 − 0.272 = 0.228, and the Gini is:

Gini = A ÷ (A + B) = 0.228 ÷ 0.5 = 0.456.

That hand-computed 0.46 lands just shy of the official figure because lumping the top fifth into one block hides the extreme concentration within it — the real top-5-percent detail would push the number up. But it is remarkably close for five data points and a bit of arithmetic, and it demystifies a statistic most people treat as magic.

What the U.S. number actually is

The Census Bureau reports a Gini index for U.S. household money income of roughly 0.49 in recent years — a level that has drifted upward over recent decades and that is high relative to other wealthy democracies. The Census tracks it in the same historical inequality tables that hold the quintile shares, and its annual income report, Income in the United States: 2023, noted that the Gini and related ratios were not significantly different between 2022 and 2023 — inequality at a high plateau rather than a fresh spike.

As with the raw quintile shares, which income you measure changes the answer. The pre-tax, pre-transfer Gini is higher; the after-tax, after-transfer Gini is lower, because the Congressional Budget Office's distributional analysis shows taxes and government transfers meaningfully compress the distribution. A headline Gini that doesn't say whether it is before or after government action is only half a fact.

Where the single number misleads

The Gini's great strength — one number for a whole distribution — is also its great weakness. Compression discards information, and three blind spots deserve respect.

First, identical Ginis can hide very different societies. A country where the middle class is squeezed and a country where the very bottom is destitute can post the same coefficient through entirely different Lorenz-curve shapes. The Gini tells you how much inequality, not where it lives. To locate it, you go back to the quintile shares or to top-share measures.

Second, it shares the underlying data's blind spots. A Gini built on Census money income inherits all the same omissions — it typically excludes capital gains (which would raise it) and non-cash transfers and benefits (which would lower it). The coefficient is only as complete as the income concept feeding it.

Third, it is a snapshot, not a movie. A single-year Gini cannot tell whether the same households occupy the bottom every year or whether people churn through it — a society with high annual inequality but lots of mobility is very different from one where the bottom is permanent, yet both can post the same Gini.

Used well, the Gini is a superb first question: it tells you instantly whether a distribution is roughly Danish, roughly American, or roughly South African, and which direction it's trending. Used badly, it becomes a verdict that papers over everything that actually matters about who is poor and why. The honest move is to treat the single number as the headline and the Lorenz curve and quintile shares as the article — read all of it before you conclude anything.