The Gini Coefficient: Measuring Inequality With One Number

The Lorenz curve: the picture the number summarizes

Start with the object the Gini actually measures. Line up everyone from poorest to richest. On the x-axis plot the cumulative share of the population (0% to 100%); on the y-axis plot the cumulative share of income they collectively hold. The result is the Lorenz curve (Max Lorenz, 1905).

If income were perfectly equal, the poorest 20% of people would earn 20% of income, the poorest 60% would earn 60%, and so on — a straight 45° diagonal called the line of perfect equality. In any real economy the poor hold less than their headcount share, so the curve sags below the diagonal: the bottom 60% of people might hold only 30% of income. The deeper the sag, the more unequal the society.

The Gini turns that sag into a number. Let A be the area between the diagonal and the Lorenz curve, and B the area under the Lorenz curve. Then:

Gini = A / (A + B)

Because the whole triangle below the diagonal has area A + B = ½, this is equivalently Gini = 2A = 1 − 2B. Perfect equality: the curve is the diagonal, A = 0, Gini = 0. Perfect inequality (one person has everything): the curve runs flat along the bottom then jumps to 100%, A fills the whole triangle, Gini → 1.

A worked example you can do by hand

Take a toy economy of 5 people with incomes 1, 2, 3, 4, 10 (total = 20). Sort them (already sorted) and build cumulative income shares:

• Poorest 20% (1 person): 1/20 = 5%
• Poorest 40%: 3/20 = 15%
• Poorest 60%: 6/20 = 30%
• Poorest 80%: 10/20 = 50%
• Everyone (100%): 20/20 = 100%

Those five points define the Lorenz curve. A fast discrete formula avoids measuring areas by hand. Sort incomes yᵢ ascending, i = 1…n:

Gini = ( 2 · ∑ i·yᵢ ) / ( n · ∑ yᵢ ) − (n + 1) / n

Here ∑ i·yᵢ = 1·1 + 2·2 + 3·3 + 4·4 + 5·10 = 1 + 4 + 9 + 16 + 50 = 80, ∑ yᵢ = 20, n = 5:

Gini = (2 · 80) / (5 · 20) − 6/5 = 160/100 − 1.2 = 1.6 − 1.2 = 0.40

So this five-person economy scores 0.40 — roughly the level of the United States. Now double the top earner to 20 (incomes 1, 2, 3, 4, 20, total 30): the same formula gives ∑ i·yᵢ = 1 + 4 + 9 + 16 + 100 = 130, and Gini = (2·130)/(5·30) − 1.2 = 260/150 − 1.2 = 1.733 − 1.2 ≈ 0.53. One rich person pulling away moved the whole number sharply — a preview of what the Gini does and doesn't see.

Why one number is both the point and the problem

The Gini's power is that it satisfies four properties economists want in an inequality index: it is (1) scale-independent — multiply everyone's income by 10 and it doesn't change, so it measures relative not absolute gaps; (2) population-independent — merging two identical countries leaves it unchanged; (3) anonymous — only the distribution of income matters, not who holds it; and (4) it obeys the Pigou–Dalton transfer principle — moving a dollar from a richer to a poorer person (without reordering them) always lowers it.

But collapsing a whole curve into one scalar means very different societies can share the same Gini. A country where the middle class is squeezed and a country where the poorest are destitute can both read 0.40. The Gini is mathematically most sensitive to changes near the middle of the distribution (where most people are) and relatively insensitive to the extreme tails. That is exactly backwards from what dominates political debate — the top 1% and outright poverty. Two Lorenz curves can even cross: one society more equal at the bottom, the other more equal at the top. When curves cross, the Gini still spits out a ranking, but that ranking is not robust — a different, equally reasonable index could flip it.

Real numbers, real countries, real history

Cross-country Gini figures (World Bank / OECD, disposable income after taxes and transfers) tell a clear story:

• Slovenia, Slovakia, Nordic states, Czechia: ≈ 0.24–0.28 — the most equal market economies on Earth, driven by heavy transfers.
• Germany, France, Canada: ≈ 0.29–0.33.
• United States: ≈ 0.39–0.41 — high for a rich country.
• Brazil, Colombia: ≈ 0.49–0.53.
• South Africa: ≈ 0.63 — the highest measured, a durable legacy of apartheid.

Two subtleties matter enormously. First, pre-tax vs post-tax: the U.S. market-income (pre-tax) Gini is ~0.51, but taxes and transfers pull the disposable-income Gini down to ~0.39 — the ~0.12 gap is the redistributive state made visible. Second, income vs wealth: wealth is always far more concentrated. The U.S. wealth Gini is roughly 0.85, because millions of households have near-zero or negative net worth while a sliver holds most assets. Historically, the U.S. income Gini fell through the mid-20th-century "Great Compression" to a low around 0.35 in the late 1960s, then rose steadily from the 1980s — a widely cited empirical anchor of the modern inequality debate.

Assumptions, pitfalls, and what the number quietly hides

Every Gini rests on choices that can move it more than the underlying reality:

• Unit of analysis: individuals vs households? Household Ginis are lower because incomes are pooled. Adjusting for household size ("equivalized" income) changes the figure again.
• What counts as income: including non-cash transfers, imputed rent, capital gains, or the government services people consume can shift a country's Gini by several points.
• The life-cycle illusion: a society of identical people who are simply at different ages — students poor, mid-career rich, retirees drawing down — shows a positive Gini even with zero lifetime inequality. A snapshot cannot see mobility. Lifetime Ginis are typically well below annual ones.
• Data truncation: household surveys systematically miss the very rich (non-response, capped top-coding), so official Ginis understate top-end inequality; tax-record studies (Piketty–Saez–Zucman) tend to read higher.
• Informal economies: in developing countries, unmeasured cash and subsistence output blur the picture.

None of this makes the Gini useless — but it means a comparison is only valid when the two Ginis were built the same way, from the same income definition and unit. Comparing a household post-tax Gini to an individual pre-tax one is a category error people make constantly.

The misconception: Gini is not a percentage of anything

The single most common error is treating the Gini as a share — reading 0.40 as "the top 40% own everything" or "40% of income is unequal." It is neither. It is a normalized area ratio with no direct real-world referent; there is no group whose income share equals the Gini. A cleaner intuition: the Gini equals half the average absolute income difference between two randomly chosen people, expressed as a fraction of mean income. Formally Gini = (mean absolute difference) / (2 × mean income). So a Gini of 0.40 means that if you pick two people at random, the expected gap between their incomes is about 80% of the mean income (2 × 0.40). That is a concrete, checkable statement — unlike the phantom "40%."

A second subtle point: a rising Gini does not automatically mean the poor got poorer. Because the measure is purely relative, everyone's real income can grow while the Gini rises — if the rich grew faster. Conversely a recession that hammers the top can lower the Gini while making almost everyone worse off. The Gini answers "how spread out?", never "how well off?" — which is why it should always be read alongside a level statistic like median income or a poverty rate.

Gini coefficient vs. other inequality measures

Measure	What it captures	Strength	Weakness
Gini coefficient	Overall dispersion; area under Lorenz curve	One intuitive number, 0–1, comparable across countries	Insensitive to WHERE inequality happens; hides distribution shape
Palma ratio	Income of top 10% ÷ bottom 40%	Directly tracks the extremes that dominate politics	Ignores the entire middle of the distribution
Theil index (T)	Entropy-based; ∑ shares × log(shares)	Decomposable into within-group + between-group	Not bounded to a clean 0–1; less intuitive
90/10 or 80/20 ratio	Ratio of a high percentile to a low one	Trivial to explain and compute	Discards everyone between the two percentiles
Top 1% income share	Fraction of income going to the richest 1%	Captures the tail Gini smoothes over	Says nothing about the poor or the middle