The Lorenz curve and the Gini coefficient: how to draw, calculate and read them

Read enough past papers and the inequality questions sort themselves. Knowing which kind you are facing tells you which tool to reach for, and most can be answered in under a minute once you do.

This guide stands on its own, and is the place for inequality in this resource set: the basic statistics module covers describing data, and inequality will get a development-economics module of its own. The Delhi School’s harder versions of these questions are in the practice set at the end.

A Lorenz curve answers one question: what share of the total income do the poorest x% of households hold? Six steps draw it, by hand or in a spreadsheet:

1. Sort the incomes from the smallest to the largest. Skipping this step is the commonest error.
2. Share: divide each income by the total.
3. Accumulate the income shares from the poorest upwards. The last one must be 100%.
4. Accumulate the household shares. With n equal households each adds 1/n. With groups of different sizes, add each group’s share of the head-count.
5. Plot the cumulative income share (vertical) against the cumulative household share (horizontal), starting from the origin (0, 0). Use a square so the diagonal really is at 45°.
6. Join the points, and draw the line of equality from (0, 0) to (100%, 100%) for comparison.

Try it. The lab holds ten households in a ward of Ashanagar (invented incomes, in ₹ thousand a month). Edit any income, add or remove a household, or pick one of the examples, and watch the table, the curve and three calculations of the Gini update together.

For the default ten households, which total ₹200 thousand, the cumulative income shares come out as 3%, 7%, 12%, 18%, 25%, 33%, 43%, 55%, 70% and 100% for the poorest 10%, 20% and so on up to 100%. The poorest half, five households, hold 25% of the income, and the richest household alone holds 30%. Three properties are worth remembering because exams ask them as definitions:

• The curve starts at (0, 0), ends at (100%, 100%), and never rises above the diagonal, since the poorest x% cannot hold more than x%.
• It bends upward more steeply as it goes, because each new household is richer than the last.
• If every household had the same income the curve would be the diagonal itself, the 45° line (Hyderabad 2014). The Lorenz curve measures inequality, and nothing else, not welfare, poverty or development (Hyderabad 2016).

Reading a cumulative share from a table

The JNU 2017 question gives income shares for five groups of 20% each, 7.5%, 11.8%, 15.5%, 21.1% and 44.1%, and asks for the cumulative share of the bottom 60%. It is the third step alone: 7.5 + 11.8 + 15.5 = 34.8%. The bottom 80% hold 55.9%, which is also 100 − 44.1.

Mistakes that cost marks

• Plotting each group’s own share instead of the cumulative share. The curve then looks like a staircase going nowhere near (100%, 100%).
• Forgetting that the horizontal axis is also cumulative, and labelling it with the group’s income instead of the share of households.
• Leaving out the origin, or drawing the curve above the diagonal. Above the diagonal means the poor hold more than their head-count share, which cannot happen after sorting.

Call A the area between the line of equality and the curve, and B the area under the curve. Together they make a triangle of area ½, so the Gini coefficient G = A ÷ (A + B) = 2A = 1 − 2B. Four routes to the same number follow. The lab above shows the first three side by side for any incomes you enter.

Route 1: the area under the curve

The curve between consecutive points is a straight line, so the area under each stretch is a trapezoid: its width times the average of its two heights. Add them up to get B.

For the ten households each Δp is 0.1, and the trapezoids are 0.0015, 0.005, 0.0095, 0.015, 0.0215, 0.029, 0.038, 0.049, 0.0625 and 0.085. They add to B = 0.316, so G = 1 − 0.632 = 0.368.

Route 2: the gaps between every pair

The Gini is also the average gap between two households, as a share of twice the mean income. No curve is needed.

The gaps for the ten households add up to 736 over unordered pairs, so 1,472 over all ordered pairs. With n = 10 and a mean of 20, G = 1,472 ÷ (2 × 100 × 20) = 0.368. For a society where half the people earn ₹100 and half earn ₹200, a pair is unequal half the time, with a gap of 100, so the average gap is 50 and G = 50 ÷ (2 × 150) = 1/6, the answer to the DSE 2012 question.

Route 3: the rank formula

Sort the incomes from the smallest, y₁ ≤ y₂ ≤ … ≤ y_n, and weight each by its rank i. This formula is the one UGC NET asked for in 2023.

For the ten households Σ i·y_i = 6 + 16 + 30 + 48 + 70 + 96 + 140 + 192 + 270 + 600 = 1,468. Then G = 2 × 1,468 ÷ (100 × 20) − 11 ÷ 10 = 1.468 − 1.1 = 0.368. Try four incomes, 1, 3, 6 and 10, with mean 5: Σ i·y_i = 65, so G = 130 ÷ 80 − 5/4 = 1.625 − 1.25 = 0.375, and the pairwise route gives 30 ÷ 80 = 0.375 too.

One note on the paper itself. UGC NET 2023 keys option B, this formula. Option A, a single sum of the gaps y_i − y_j over j ≤ i divided by n²μ, gives exactly the same number once the incomes are sorted. It adds each pair’s gap once, where Route 2’s double sum adds it twice and divides by twice as much. So it is not wrong either; if you ever see this question, choose B and know why A fits as well. Option D forgets the factor 2 and gives twice the Gini.

Route 4: from a Lorenz function

If the Lorenz curve is a formula L(p), the area is an integral and G = 1 − 2∫₀¹ L(p) dp. For L(p) = p^k, ∫ is 1 ÷ (k + 1), so G = 1 − 2 ÷ (k + 1).

Published tables rarely list households. They give the income shares of five groups (quintiles) or ten groups (deciles). Plot the cumulative points and join them with straight lines, then use the trapezoid route on the polygon. For k equal groups, with cumulative shares L_i:

Enter any five shares below, or start with the JNU 2017 table for Niger, which gives a grouped Gini of 0.33. The key fact is the second preset: for the sixty Ashanagar households, the true Gini from individual incomes is 0.256 but the quintile shares give only 0.242.

The next two kinds of question are about reading, not calculating. Sensible measures of inequality obey four rules: anonymity (who earns what does not matter), scale independence (multiplying every income changes nothing), population independence (copying the whole population changes nothing) and the transfer principle (a rupee moved from a richer to a poorer household, without reversing their order, lowers inequality). The Gini obeys all four.

JNU 2010 tests the first three with six distributions of four kinds of people. Tick distributions to put them on the plot. D2 reorders the incomes, D3 multiplies them by five, D6 has five of every kind of person. Look at the Gini column, the standard deviation column and the curves.

The Gini of D1, D2, D3 and D6 is 0.25 in every case, so G₅ < G₁ = G₂ = G₃ = G₆ < G₄, the keyed answer (JNU 2010). The standard deviation tells a different story, σ₅ < σ₁ = σ₂ = σ₆ < σ₄ < σ₃, because multiplying every income by five multiplies the spread by five (the follow-up question in the same paper). That contrast is the point: the Gini is a relativemeasure, and the standard deviation is not. JNU 2021 gives two distributions of incomes 100, 200, 300, 400 and 200, 400, 600, 800 with the same head-counts. The second is the first doubled, so the Lorenz curves are the same.

When two curves cross

Take two societies of ten people. In A, seven earn ₹1,000 and three earn ₹2,000. In B, five earn ₹1,000 and five earn ₹2,000 (DSE 2013). Slide along the curves to see where the lead changes hands.

A is ahead for the poorest half, and B takes the lead somewhere between the fiftieth and sixtieth percentile. The Gini coefficients, 0.1615 and 0.1667, put them in an order only by averaging away the disagreement. The right answer to “which has less inequality?” when Lorenz curves cross is that we cannot say (JNU 2015, JNU 2016). If one curve lies nowhere below the other and above it somewhere, the upper curve is less unequal by every measure that obeys the transfer principle.

A cousin of this question uses distribution functions instead of Lorenz curves. If the distribution function of expenditure for 2001 lies everywhere to the right of that for 1991, mean expenditure is higher. Nothing follows about every individual, or about the Gini, which can move either way (DSE 2012).

Simple shares and ratios

A single number hides where in the distribution the inequality sits. Reports therefore quote shares beside it: the share of the richest 10%, the share of the poorest 40% or 50%, and ratios built from them. The Palma ratio divides the share of income held by the richest 10% by the share held by the poorest 40%. It is a ratio of two points on the Lorenz curve. For the sixty Ashanagar households it is 19.4% ÷ 23.5% = 0.83, so the top tenth hold less than the bottom four-tenths. For the ten households of the lab it is 30% ÷ 18% = 1.67. A Palma above 1 means the top tenth out-earn the bottom four-tenths together.

Inequality and development: Kuznets

Simon Kuznets proposed that inequality first rises and then falls as an economy develops, an inverted U between income per head and inequality. The Kuznets curve is the relationship between a country’s income per capita and the inequality of its income distribution (CUET 2024, GATE 2021). On the evidence, studies using many countries at one moment tend to support the inverted U more than studies following one country through time (JNU 2020).

JNU 2016 builds the idea from five workers who all begin in a traditional sector at a wage of 1. In each period one moves to a modern sector that pays 4. Move the slider.

The Gini goes 0, 0.30, 0.33, 0.26, 0.14, 0: it at first rises and then falls, the keyed answer. Inequality rises as long as the modern sector is small, because the few who have moved pull away from the many who have not, and falls once most have joined. The mean wage rises the whole time.

A related statement question (JNU 2017) is worth the minute. “If the incomes of some people grow faster than others, inequality rises” is only sometimes true: if the poor’s incomes grow faster in proportion, inequality falls. “If inequality rises, the incomes of some people grow faster than others” is always true, since if every income grew at the same rate the shares would not change. The keyed answer is that I is sometimes true and II is always true.

In 2025 headlines announced that India had become one of the most equal societies in the world, on the strength of a Gini of 25.5. The World Bank’s own database holds the number, and the survey behind it. It is a good lesson in reading a Gini. All the figures below come from the World Bank’s Poverty and Inequality Platform and World Development Indicators (sources at the end); the Gini is on the 0 to 100 scale the Bank uses.

India’s Gini was 29.7 in 1977, between 26 and 29 through the following decades, 28.8 in 2011 and 25.5 in 2022. Three things about the series are worth noticing.

1. Every point is a consumption Gini. The surveys ask what households spend, not what they earn. Spending is smoother than income, because households save in good times and borrow in bad ones, and richer households save a larger share of what they earn. For this reason a consumption Gini is usually lower than an income Gini for the same country.
2. The survey changed. The 2022 figure comes from a different survey, the Household Consumption Expenditure Survey, from the National Sample Survey rounds before it, and the platform places it in a different comparability group from 2011. The fall from 28.8 to 25.5 is therefore not a like-for-like change. The World Bank’s own October 2022 brief for India gave a Gini of 35.7 for 2019, built from consumption imputed from a private survey, about ten points above the 2022 figure. The years and the data differ, so it is not a measured change, but it shows how much the source matters.
3. Countries are measured differently. The ranking in the press placed India fourth, behind the Slovak Republic, Slovenia and Belarus. The next chart shows why that comparison is loose.

The Slovak Republic (23.8) and Slovenia (24.7) are income Ginis; Belarus (24.4) and India (25.5) are consumption Ginis. Setting an income Gini beside a consumption Gini is not like for like. Many high-income countries are measured on income and many developing countries on consumption, so a league table mixes two different yardsticks.

What a consumption Gini misses

A survey of household spending hardly reaches the very rich, and what they spend is a small part of what they earn. A different kind of evidence, income-share studies that combine tax records, surveys and national accounts, looks at the top directly. The World Inequality Lab study by Bharti, Chancel, Piketty and Somanchi (March 2024) reports that by 2022-23 the richest 1% of Indians received 22.6% of national income and held 40.1% of wealth, which the authors describe as the highest levels in the period they cover, 1922 to 2023. That is a share, not a Gini, and it comes from different data and methods. It does not contradict a consumption Gini of 25.5. The two answer different questions.