Common mistake 1 with Lorenz curve and Gini

Comparing a Gini across sources without checking what was measured. Pre-tax against post-tax, income against consumption, household against individual - each shifts the number by more than most of the differences people argue about.

Common mistake 2 with Lorenz curve and Gini

Reading a change in the Gini as a change at any particular point in the distribution. It is one summary of a whole curve, and it can stay flat while the bottom and the top both move.

Common mistake 3 with Lorenz curve and Gini

Using it on a variable that can be negative, or on counts of something already averaged.

Common mistake 4 with Lorenz curve and Gini

Comparing Ginis across countries or years computed from different income concepts or units. The differences between definitions routinely exceed the differences being reported.

EconStatsCraft › Lorenz curve and Gini

Lorenz curve and Gini

How unevenly something is shared. The curve plots the cumulative share held by the poorest p of the population; the Gini is twice the area between that curve and the 45-degree line of perfect equality. Computed from the points that are drawn, so the number and the picture cannot disagree.

What it needs

What is shared out - Income, wealth, land, emissions - anything non-negative.

The intuition

Sort everyone from poorest to richest and plot the cumulative share of people against the cumulative share of income. Perfect equality is the 45-degree line: the poorest 30% would hold 30% of the income. Real distributions sag below it, and the Gini coefficient is simply how much area lies in that sag, doubled so it runs from 0 to 1. One number for the whole curve - which is its convenience and its central weakness.

When to use it

For any question about how unevenly something is shared: income, wealth, land, firm size, emissions, market share. Also as a summary of concentration in industrial organisation, where the same curve is read as market share by firm.

Where the estimator comes from

With incomes sorted x_{(1)}\le\dots\le x_{(n)}, the Lorenz curve joins the points \left(\frac{k}{n},\ \frac{\sum_{i\le k}x_{(i)}}{\sum_i x_i}\right). The Gini is twice the area between that curve and the diagonal, and has several equivalent forms. The most revealing is the mean absolute difference form, G=\frac{\sum_i\sum_j|x_i-x_j|}{2n^2\bar{x}}, which says the Gini is the average absolute income gap between two randomly chosen people, expressed as a share of twice the mean. The computational form used here is G=\frac{2\sum_i i\,x_{(i)}}{n\sum_i x_i}-\frac{n+1}{n}, which is one pass after sorting.

The model

G=1-2\int_0^1 L(p)\,dp\qquad L(p)=\frac{\sum_{i\le p}y_i}{\sum_i y_i}

What it assumes

The variable is non-negative. Negative income - a loss-making firm, a household in debt - breaks the construction, because a cumulative share can then fall as you move up the distribution.
Every unit is one unit. If the rows are households of different sizes, a household Gini and a person Gini are different numbers and the household one usually understates inequality. Weight or expand first.
The sample represents the population. Survey data with sampling weights should be weighted; this chart does not weight, so use it on already-representative data or on a collapsed table.
The unit of analysis is the one the question is about. Inequality across individuals, households and households adjusted for size give materially different Ginis from the same survey.
The income concept is stated. Pre-tax, post-tax and post-transfer inequality differ by more than most policy changes, and a Gini without that label is not interpretable.

Statistical properties

The sample Gini cannot reach 1. With one person holding everything among n, it equals 1 - 1/n: 0.99 for n = 100. The small-sample bias is downward and a correction of n/(n-1) is standard.
Very different distributions can share a Gini. The incomes [1,1,1,1,6] and [1,1,2,7,7] both give exactly 0.4000, so the number identifies neither the shape of the distribution nor where the inequality sits.
The Gini is most sensitive to changes around the MIDDLE of the distribution and relatively insensitive to the extreme tails, which is the opposite of what most inequality debates are about. Top-income shares or the Theil index respond where the Gini does not.
It is scale-invariant - doubling every income leaves it unchanged - and satisfies the Pigou-Dalton transfer principle: any transfer from richer to poorer lowers it.
It is not decomposable by subgroup in a clean way. Between-group and within-group components do not sum to the total unless the groups do not overlap, and the residual term has no natural interpretation. The Theil index decomposes exactly and is preferred for that purpose.
Lorenz curves that CROSS cannot be ranked by any inequality measure without imposing extra value judgements. Two distributions with the same Gini and crossing curves are genuinely not comparable.

How to read the output

The Gini is one number for a whole curve, so two very different distributions can share it. Look at the curve as well: one that sags mostly at the bottom is a poverty story, one that only departs at the top is a concentration-at-the-top story, and both can give 0.4.
Roughly: 0.25-0.30 is a Nordic income distribution, 0.35-0.40 is much of Europe, 0.45-0.50 is Brazil or South Africa territory. Wealth Ginis run far higher than income Ginis for the same country - often above 0.7 - so never compare one with the other.
The Gini here is computed as the area under the curve that is drawn, not by a separate formula, so if the picture looks wrong the number is wrong in the same way.
The Gini is most sensitive to changes around the middle of the distribution and least sensitive at the extremes, which is the opposite of where most policy interest lies.
A Lorenz curve that lies entirely inside another means unambiguously less inequality. When two curves cross, no single index can rank them without a value judgement about which part of the distribution matters.

A worked reading

On the wage sample the Gini is 0.2542 - low by international standards, which is what a single occupation-and-schooling dataset looks like rather than a national income distribution. Then the warning that the number alone cannot give you. Take the two small distributions [1,1,1,1,6] and [1,1,2,7,7]: both return a Gini of 0.4000, and they are not remotely the same society. One is four identical people and a single rich outlier; the other has two rich, one in the middle and two poor. Any policy aimed at the first would miss the second entirely. The Gini is a single summary of an entire curve, and every single summary of a curve has a family of distributions that share it - which is why the Lorenz curve is drawn beside the coefficient rather than under it, and why a change in the Gini over time should always be read off the two curves before it is explained.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Gini above 0.9	Near-total concentration, or a data error	Check for a single enormous value
Gini near zero on real income data	Almost certainly the wrong variable	Real income Ginis run roughly 0.25 to 0.65
Negative incomes present	The Lorenz curve is not defined	Losses break the cumulative share; exclude or bottom-code them
Two groups, same Gini, different curves	The curves cross	They cannot be ranked. Show both curves
Gini barely moving after a large transfer	It was in a tail	Use top shares or Theil for tail-sensitive questions
Small sample	The maximum is 1 - 1/n	Apply the n/(n-1) correction

Common mistakes

Comparing a Gini across sources without checking what was measured. Pre-tax against post-tax, income against consumption, household against individual - each shifts the number by more than most of the differences people argue about.
Reading a change in the Gini as a change at any particular point in the distribution. It is one summary of a whole curve, and it can stay flat while the bottom and the top both move.
Using it on a variable that can be negative, or on counts of something already averaged.
Comparing Ginis across countries or years computed from different income concepts or units. The differences between definitions routinely exceed the differences being reported.

Where to read this properly

Gini (1912), Variabilita e Mutabilita.
Lorenz (1905), "Methods of Measuring the Concentration of Wealth", Publications of the ASA 9(70).
Cowell, Measuring Inequality (2011), ch. 2 and 3.
Atkinson (1970), "On the Measurement of Inequality", Journal of Economic Theory 2(3), on the value judgements involved.

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Lorenz curve and Gini

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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