The Lorenz Curve & Gini Coefficient

The Lineup

Imagine lining up every person in a country, from the poorest to the richest, shoulder to shoulder. Now ask: what share of the total national income does the bottom fifth take home? What about the bottom half?

This is the kind of question that sounds simple until you try to answer it, and then sounds devastating once you do. In the United States, the bottom 20% of earners take home roughly 3% of total income.1 The bottom 50% earn about 12%. Which means the top 10% take home nearly as much as everyone else combined.

These numbers are arresting, but they're also just snapshots — individual data points ripped from what is really a continuous curve. In 1905, a young PhD student at the University of Wisconsin named Max Otto Lorenz had the insight that you could tell the whole story at once, in a single picture.2 He was twenty-nine years old. The picture he drew would become one of the most reproduced diagrams in all of economics.

Here's how it works. You've already lined everyone up from poorest to richest. Now, walking down the line from left to right, you keep a running total: after passing 10% of the population, how much of the total income have you accounted for? After 20%? After 30%? Plot those cumulative shares on a graph — population percentage on the horizontal axis, cumulative income share on the vertical — and you get Lorenz's curve.

If incomes were perfectly equal — every person earning exactly the same — then after passing 20% of the population, you'd have accounted for 20% of the income. After 50%, exactly 50%. The Lorenz curve would be a straight diagonal line at 45 degrees. Economists call this the line of equality.

If incomes were as unequal as mathematically possible — one person owns everything, everyone else earns zero — then the curve would hug the bottom axis all the way until the very last person, where it would shoot straight up to 100%. That's the line of perfect inequality.

Every real society falls somewhere in between, with its Lorenz curve bowing below the diagonal. The deeper the bow, the more unequal the society. Sweden's curve hugs the diagonal relatively closely. South Africa's sags dramatically. The United States is, as in many things, somewhere in the anxious middle.

One Number to Rule Them All

Lorenz's curve is beautiful and informative, but it has a practical problem: you can't rank countries by a curve. You can't plug a curve into a regression. Policy makers want a number. And in 1912, the Italian statistician Corrado Gini gave them one.3

The idea is wonderfully geometric. Look at the area between the line of equality and the Lorenz curve — the shaded region in our diagram, call it A. Now look at the entire triangle below the line of equality — call that A + B (where B is the area under the Lorenz curve). The Gini coefficient is simply:

Since the total triangle always has area 0.5 (it's a right triangle with legs of length 1), this simplifies to G = 2A, or equivalently, G = 1 − 2B. The Gini coefficient ranges from 0 (perfect equality) to 1 (one person has everything).4

And now the world can be ranked. As of recent World Bank estimates: Sweden sits at about 0.27. Germany at 0.32. The United States at 0.39. Brazil at 0.49. South Africa, one of the most unequal nations on Earth, at 0.63.5

There's an elegant alternative formulation that doesn't require drawing any curves at all. Pick two people at random from the population. The Gini coefficient equals the expected absolute difference in their incomes, divided by twice the mean income. This is the "mean absolute difference" definition, and it's beautiful precisely because it shows what the Gini is really measuring: how different two randomly chosen people are likely to be.

Build Your Own Inequality

The best way to understand the Lorenz curve is to build one. Below, you can set incomes for ten people and watch the curve and Gini coefficient respond in real time. Try the presets to see how different countries look, or create your own dystopia.

Notice how small changes at the top of the distribution have an outsized effect on the Gini. Give the richest person a raise and the curve sags dramatically. Give the poorest person a raise and… not much happens. This asymmetry is baked into the geometry — the Gini is more sensitive to transfers in the middle of the distribution than at the tails, a fact that has important policy implications.6

The Gini's Dirty Secret

Here is the thing they don't tell you in the summary statistics: two societies can have exactly the same Gini coefficient and look absolutely nothing alike.

Consider two fictional countries. In Midlandia, there's a comfortable middle class and moderate extremes — the rich are well-off but not obscenely so, the poor are struggling but not destitute. In Bimodia, there's a tiny ultra-wealthy elite and a large impoverished underclass, with almost no one in between. These two countries can have identical Gini coefficients.

How is this possible? Because the Gini compresses an entire curve — an infinite amount of information — into a single number. Different curves can enclose the same area. It's like saying two lakes have the same surface area: one might be a long, thin fjord, the other a near-perfect circle. Same number, completely different geography.

Statisticians call this the problem of crossing Lorenz curves. When one curve is everywhere below another, we can unambiguously say the first represents more inequality — this is called Lorenz dominance. But when curves cross, the ranking depends on which part of the distribution you care about most. The Gini, by weighting the middle most heavily, effectively makes that choice for you — without asking.

Explore this for yourself below. Two distributions, same Gini, different worlds.

Beyond Income

One of the most surprising things about the Gini coefficient is how far it travels from its home in economics. The same math that measures income inequality can measure any kind of distributional unevenness.

Health researchers use the Gini to measure inequality in life expectancy across regions.7 Education scholars apply it to years of schooling. Ecologists have adapted it to measure biodiversity — how evenly distributed are species across an ecosystem? A forest dominated by one species has a high "ecological Gini"; a diverse rainforest has a low one.

Land ownership Ginis are among the most striking. In many Latin American countries, the land Gini exceeds 0.8 — far higher than income Ginis — reflecting centuries of colonial land grants that concentrated vast estates in a few families. Paraguay's land Gini is estimated at 0.93.8 Essentially, a handful of families own the country's arable land.

There's also an intimate connection to the power-law distributions we explored in Chapter 94. Vilfredo Pareto — another Italian, and Gini's near-contemporary — observed that wealth distributions follow a particular mathematical shape, where the probability of having income greater than x falls off like x^−α. The famous "80/20 rule" (20% of people hold 80% of wealth) is a consequence of a Pareto distribution with a specific exponent. And it turns out you can express the Gini coefficient directly in terms of Pareto's exponent α:

This formula is a gem. It tells you that in a Pareto world, the Gini only depends on the tail exponent — the fatness of the tail. Inequality, in this framework, is entirely determined by how extreme the extremes are. And in many empirical wealth distributions, Pareto's law holds surprisingly well in the upper tail, which is precisely where inequality lives.

What the Number Hides

Let's return to the fundamental tension. The Gini coefficient is popular because it does something seemingly impossible: it compresses the entire income distribution into a number between 0 and 1. You can compare countries, track trends over time, set policy targets. The World Bank publishes Ginis for nearly every nation on Earth.

But every act of compression is also an act of erasure. Here are some things the Gini cannot tell you:

Where the inequality lives. A Gini of 0.40 could mean the rich are pulling away from everyone else, or that the poor are falling behind the middle class, or that the middle class is hollowing out in both directions. These are politically and morally very different situations.

Whether inequality is static or dynamic. A society where the same families are rich generation after generation has a very different character from one where there's high mobility — people moving up and down the ladder over their lifetimes — even if the snapshot Gini is identical.

What's happening at the extremes. The Gini is most sensitive to changes in the middle of the distribution. Billionaires could double their wealth and the Gini would barely budge, because they're such a tiny fraction of the population. Yet this is precisely the kind of inequality that dominates public discourse.

This is why serious inequality researchers rarely stop at the Gini. They look at the Palma ratio (the share of the top 10% divided by the bottom 40%), or the Theil index (which can be decomposed between and within groups), or simply the share of income going to the top 1%. Each of these captures something the Gini misses.

But none of them has the Gini's elegance. None of them maps so cleanly onto a geometric picture. And none of them is so easy to explain in a sentence: the Gini coefficient measures how far a society is from perfect equality, on a scale from zero to one. This simplicity is the Gini's greatest strength and its greatest limitation — two things that turn out, in the mathematics of summarization, to be exactly the same thing.

The Moral of the Coefficient

Max Lorenz never became famous. He spent most of his career at the Bureau of Labor Statistics, quietly working on statistical methods.2 Corrado Gini, by contrast, became one of Italy's most prominent statisticians — and, less admirably, a supporter of Mussolini's regime who tried to use statistics to support fascist ideology. The history of inequality measurement is itself unequal.

But the mathematical legacy is clear. The Lorenz curve and the Gini coefficient teach us something that goes far beyond economics: summary statistics are both necessary and dangerous. We need to compress complex realities into numbers we can compare and communicate. But every number is a lossy compression. Every statistic is a deliberate act of forgetting.

The responsible thing is not to avoid summary statistics — that way lies paralysis — but to understand what each one remembers and what it forgets. The Gini remembers the overall spread but forgets the shape. It remembers the middle but forgets the tails. It captures the amount of inequality but not its character.

So the next time someone tells you that country X has a Gini coefficient of 0.35, don't just nod. Ask: What does the curve look like? Because behind every number, there's a curve, and behind every curve, there are people — lined up from poorest to richest, each with a story that no single coefficient can tell.