Lorenz Curve and Gini Coefficient Explained (With a Worked Example)

A Lorenz curve is a graph showing how the total income of a population is shared out, and the Gini coefficient is the single number that summarizes that graph. The curve plots the cumulative share of the population, ranked from poorest to richest, on the horizontal axis, against the cumulative share of total income those people collectively receive on the vertical axis. If every household earned exactly the same, the plot would be a straight 45 degree line. Real distributions sag below that line, and the size of the sag is the inequality. The Gini turns the sag into a ratio of two areas, A / (A + B), where A is the area between the equality line and the curve and B is the area underneath the curve. It runs from 0 for perfect equality to 1 for perfect inequality. This guide builds the curve axis by axis, computes a Gini from a quintile table with every step of arithmetic shown, works out what taxes and transfers do to both, and sets out what the single number hides.

Building the Lorenz curve, axis by axis

You can follow the whole construction without seeing a picture, because both axes measure the same thing: a running total that ends at 100 percent.

Step one, rank everyone. Line up the population from the lowest income to the highest. This ordering is not optional. It is what gives the curve its shape, and it is why a Lorenz curve can only bend one way.

Step two, set the horizontal axis. The x axis is the cumulative percentage of the population, counted from the poorest end. At x = 20 you are looking at the poorest fifth of households. At x = 60 you are looking at the poorest three fifths, which includes everyone already counted at 20 and 40. The axis runs from 0 to 100.

Step three, set the vertical axis. The y axis is the cumulative percentage of total income earned by exactly the group sitting to the left of you on the x axis. The value at x = 60 answers the question: of every dollar earned in this economy, what percentage went to the poorest 60 percent of households? This axis also runs from 0 to 100.

Step four, plot and join. Each pair gives one point. Joining the points from bottom left to top right traces the Lorenz curve.

Three properties fall straight out of that construction, and they are worth stating because they let you spot a wrongly drawn curve instantly.

• The curve must start at (0, 0). Zero percent of the population earns zero percent of the income, by definition.
• The curve must end at (100, 100). All of the population earns all of the income.
• The curve can never slope downward, as long as no one's income is negative. Adding another household to the group can only add income, never subtract it, so the running total on the y axis only rises. Where negative amounts are possible, most often in wealth data, the curve can dip below zero at the bottom, and that is the one case where this rule breaks.

There is a fourth property that carries real analytical content. The slope of the Lorenz curve at any point equals that household's income divided by the mean income of the whole population. At the poor end, where incomes are below average, the slope is less than 1 and the curve rises gently. At the rich end, where incomes are above average, the slope is greater than 1 and the curve rises steeply. The slope passes through exactly 1 at the percentile where a household earns precisely the mean. Because the population is ranked poorest first, the slope only ever increases as you move right, which makes the curve convex: it bows away from the equality line and toward the bottom right corner. If your sketch bows the other way, you have plotted the ranking backwards.

The line of perfect equality and the case of perfect inequality

The line of perfect equality is the diagonal from (0, 0) to (100, 100). It says that any x percent of the population receives exactly x percent of the income: the poorest 20 percent get 20 percent, the poorest half get half. It is a 45 degree line only when both axes use the same scale, which is the standard way to draw it, and that convention is what makes the two areas comparable.

Perfect inequality is the opposite extreme: one household holds all the income and everyone else has nothing. The plot then hugs the bottom of the box. It runs flat along the horizontal axis all the way from (0, 0) to (100, 0), because every group short of the very last household has zero income between them, then jumps vertically up the right hand edge to (100, 100) when that final household is included. The area under this curve is zero.

Between these two extremes sits every real distribution. The further the curve bows away from the diagonal, the more unequal the distribution. The vertical gap between the diagonal and the curve at any x is the most readable single feature: it tells you how many percentage points of total income the poorest x percent are missing compared with an equal split.

The Gini coefficient as A / (A + B)

Label the two regions inside the square:

• A is the area between the line of perfect equality and the Lorenz curve, the crescent shaped gap.
• B is the area under the Lorenz curve, between the curve and the axes.

The Gini coefficient is

Gini = A / (A + B)

A + B is simply the whole area under the equality line, which is a right triangle covering half the box. If you scale both axes to run from 0 to 1 instead of 0 to 100, the box has area 1 and that triangle has area 0.5. So the formula collapses to something much easier to use:

Gini = A / 0.5 = 2A, and since A = 0.5 minus B, Gini = 1 minus 2B.

Now the range makes sense without memorizing it. Under perfect equality the curve is the diagonal, so A = 0, and Gini = 0 / 0.5 = 0. Under perfect inequality the curve runs along the axes, so B = 0 and A takes the whole triangle, giving Gini = 0.5 / 0.5 = 1. Every real distribution lies in between, so the Gini is trapped between 0 and 1. A higher Gini means more inequality.

Two practical notes. First, a strict version of the upper bound: with a finite population of N households, the maximum possible Gini is 1 minus 1/N rather than exactly 1, because even the single rich household is one of the N. With a large population the difference is invisible. Second, many published sources multiply the figure by 100 and report a Gini index from 0 to 100. The two are the same statistic on different scales, and a Gini of 0.39 and a Gini index of 39 say the same thing. Neither one is a percentage of income, which is the mistake covered in the Gini coefficient glossary entry.

A fully worked Gini from a quintile table

Inequality data usually arrives as quintiles: the population split into five equal fifths, each holding 20 percent of households, with the income share of each fifth reported separately. Here is a table for an imaginary economy. Read it as five separate slices, not running totals.

The shares sum to 5 + 10 + 15 + 22 + 48 = 100, which is the first thing to check on any table like this.

### Step 1: cumulate the shares

The table gives each fifth's own share. The Lorenz curve needs running totals, so add as you go from the poorest end.

• Poorest 20%: 5
• Poorest 40%: 5 + 10 = 15
• Poorest 60%: 15 + 15 = 30
• Poorest 80%: 30 + 22 = 52
• Everyone: 52 + 48 = 100

### Step 2: write down the plotted points

Compare each point with the equality line to see the sag. At x = 80 an equal distribution would give y = 80, and this economy gives 52, so the poorest 80 percent are short by 80 minus 52 = 28 percentage points of total income. Do the same at every point and you get the vertical gaps 20 minus 5 = 15, 40 minus 15 = 25, 60 minus 30 = 30, and 80 minus 52 = 28. The widest gap here is at x = 60, and the gaps narrow again after it because the curve has to climb back to (100, 100). That is a general result rather than a coincidence: the maximum vertical gap falls exactly where the Lorenz curve's slope equals 1, which is the percentile at which a household earns the mean income.

### Step 3: measure B, the area under the curve

Switch to the 0 to 1 scale so the whole box has area 1. The y values become 0, 0.05, 0.15, 0.30, 0.52 and 1.00, and each quintile is a horizontal step of width 0.2.

Treat each segment as a trapezoid, whose area is width times the average of its two heights.

Add them: 0.005 + 0.020 = 0.025, plus 0.045 = 0.070, plus 0.082 = 0.152, plus 0.152 = 0.304.

So B = 0.304.

### Step 4: measure A

The triangle under the equality line has area 0.5, and A is whatever the curve fails to cover:

A = 0.5 minus 0.304 = 0.196

### Step 5: divide

Gini = A / (A + B) = 0.196 / (0.196 + 0.304) = 0.196 / 0.5 = 0.392

Check the denominator: 0.196 + 0.304 = 0.500, exactly the half box it should be. If your A and B do not add to 0.5, you have made an arithmetic slip, and that check catches it before the answer does.

### The one line shortcut

Because A + B is always 0.5, you can skip step 4 entirely:

Gini = 1 minus 2B = 1 minus (2 x 0.304) = 1 minus 0.608 = 0.392

Same answer, half the work. Written out fully for quintile data, the shortcut is Gini = 1 minus 0.2 x (the sum of every pair of neighboring cumulative shares). Here that sum is (0 + 0.05) + (0.05 + 0.15) + (0.15 + 0.30) + (0.30 + 0.52) + (0.52 + 1.00) = 3.04, and 1 minus (0.2 x 3.04) = 1 minus 0.608 = 0.392.

### Why this answer is slightly too low

The trapezoid method draws straight lines between the five plotted points, but the true Lorenz curve is convex and sits below those straight chords. Using the chords therefore overstates the curve's height, which overstates B, which understates A, so the grouped estimate understates the true Gini. Every Gini computed from quintile bands has this downward bias. It shrinks as you move from quintiles to deciles to percentiles, and statistical agencies working from full microdata avoid it entirely. On an exam the trapezoid answer is the expected answer, but knowing which way the error points is the kind of detail that separates a good response from a correct one.

How to read and compare two Lorenz curves

Put two curves in the same box, either two countries in the same year or one country at two dates. Two things can happen.

Case one: one curve lies entirely inside the other. If curve P is everywhere at or above curve Q, then at every point the poorer share of P's population holds at least as much of the income, and provided the curves differ somewhere, P is unambiguously the more equal distribution. This is called Lorenz dominance, and it is a strong result: when one curve dominates another, the Gini agrees, and so does every other standard relative inequality measure, meaning one that judges shares rather than absolute money gaps, that respects the principle that moving income from a richer household to a poorer one without swapping their ranks should reduce measured inequality. When curves do not cross, you do not even need a number.

Case two: the curves cross. Now one distribution is more equal at the bottom and the other is more equal at the top, and there is no ranking that all measures agree on. The Gini will still spit out two numbers and one will be larger, but that ordering is a property of how the Gini happens to weight different parts of the distribution rather than an objective fact about which society is more unequal. Crossing curves are the single strongest reason to look at the picture and not just the coefficient.

A few reading habits worth building. Check the vertical gap at 20, 50 and 80 percent to describe the shape in words: a curve that hugs the diagonal at the bottom then falls away sharply near the right has a comfortable majority and a very rich top. Check where the curves cross, because that tells you which part of the population the two distributions actually differ over. And check that both curves use the same definition of income and the same unit of analysis, because a market income curve for individuals and a disposable income curve for households are not comparable no matter how neatly they sit in the same box. General graph reading habits like these carry across the subject, and the how to read economics graphs guide covers the rest of them.

Taxes and transfers: the pre-tax and post-tax Gini

Governments redistribute, and the Lorenz curve is the clearest way to see it. Agencies that publish inequality statistics typically report two Ginis for each country: one on market income, which is what households earn before the state gets involved, and one on disposable income, which is market income minus direct taxes plus cash transfers. The gap between the two is a measure of how much redistribution the tax and transfer system actually does.

The mechanics are straightforward once you think in shares.

• A progressive tax takes a larger percentage of income from higher earners, so it cuts the share of the post-tax total going to the highest earners and raises the share going to those taxed at below the economy-wide average rate. It pulls the curve up toward the diagonal and lowers the Gini.
• A transfer payment, such as a pension, unemployment benefit or child allowance, adds income at the bottom, which raises the bottom quintiles' shares directly. It also pulls the curve up.
• A proportional tax with no allowances changes nothing at all. If every income is multiplied by the same (1 minus t), every share stays exactly what it was, so the curve and the Gini are unmoved.
• A regressive tax takes a larger percentage from lower earners, pushing the curve away from the diagonal and raising the Gini.

Take the same imaginary economy and apply a tax and transfer system. The post-tax, post-transfer shares are now:

The new shares sum to 8 + 13 + 17 + 22 + 40 = 100. Cumulating: 8, then 8 + 13 = 21, then 21 + 17 = 38, then 38 + 22 = 60, then 60 + 40 = 100. On the 0 to 1 scale the heights are 0.08, 0.21, 0.38, 0.60 and 1.00.

Trapezoids again, each of width 0.2:

• 0.2 x (0 + 0.08) / 2 = 0.008
• 0.2 x (0.08 + 0.21) / 2 = 0.029
• 0.2 x (0.21 + 0.38) / 2 = 0.059
• 0.2 x (0.38 + 0.60) / 2 = 0.098
• 0.2 x (0.60 + 1.00) / 2 = 0.160

Total B = 0.008 + 0.029 + 0.059 + 0.098 + 0.160 = 0.354.

Then A = 0.5 minus 0.354 = 0.146, and

Gini = 0.146 / 0.5 = 0.292

Or by the shortcut, 1 minus (2 x 0.354) = 1 minus 0.708 = 0.292. The two methods agree, as they must.

So redistribution moved the Gini from 0.392 to 0.292, a fall of 0.100, which is just over a quarter of the original figure (0.100 / 0.392 = 0.255, or 25.5 percent). The whole post-tax curve lies above the pre-tax curve at every point, which means this is a case of Lorenz dominance and no measure would disagree about the direction of the change.

Two honest caveats about these published comparisons. In kind transfers, such as public health care and schooling, are usually left out of the disposable income measure even though they are worth a great deal to lower income households, so the standard figures understate how much redistribution occurs. And measured redistribution says nothing about whether it was worth doing, since taxes and transfers change incentives as well as shares, which is the efficiency and equity trade off that runs through the whole of the market failure and government topic. The mechanics of who actually bears a tax are worked through in the progressive vs regressive taxes guide.

What the Gini hides

The Gini is one number standing in for a whole distribution, and compression on that scale always costs information. Four losses matter.

### A single number hides the shape

Two genuinely different societies can post the identical Gini. Compare the economy above with a second one:

Economy Y cumulates to 8, 17, 31, 46, 100. Its trapezoid areas are 0.2 x (0 + 0.08) / 2 = 0.008, then 0.2 x (0.08 + 0.17) / 2 = 0.025, then 0.2 x (0.17 + 0.31) / 2 = 0.048, then 0.2 x (0.31 + 0.46) / 2 = 0.077, then 0.2 x (0.46 + 1.00) / 2 = 0.146. Those sum to 0.008 + 0.025 + 0.048 + 0.077 + 0.146 = 0.304, exactly the same B as Economy X, so Y's Gini is also 0.392.

The two are not remotely the same place. Y's poorest fifth receives 8 percent of income against X's 5 percent, so the very bottom is better off in relative terms. But Y's richest fifth takes 54 percent against X's 48 percent, and Y's fourth quintile has been squeezed down to 15 percent, barely above the third. In plain terms, Y has a less destitute bottom and a much thinner upper middle, with more concentrated income at the very top. Their Lorenz curves cross between the 60th and 80th percentile: Y sits above X at 20, 40 and 60, then dips below at 80. One number cannot tell you any of that, which is the whole argument for plotting the Lorenz curve alongside the coefficient rather than instead of it.

### It says nothing about mobility

The Gini is a snapshot of one moment. It counts how income is spread across positions, not who occupies them or for how long. An economy where the same families sit in the bottom quintile for three generations and an economy where households churn through every quintile over a working life can report the same Gini in any given year. Static inequality and the chance of moving are separate questions requiring separate longitudinal data, and treating a Gini as evidence about opportunity is one of the most common misuses of the statistic in essays.

### It says nothing about absolute living standards

The Gini is scale invariant: multiply every income in the economy by ten and the shares are untouched, so the curve and the coefficient do not move at all. A poor country where everyone is equally near subsistence can score a lower Gini than a rich country with a comfortable poor. Inequality and poverty are different measurements answering different questions, which is exactly why absolute poverty and relative poverty are defined separately, and why any serious description of a country pairs a distribution measure with a level measure such as GDP per capita or a poverty headcount.

### It is sensitive to definitions and to survey error

Before comparing two Ginis, check they measure the same thing. Income or consumption or wealth? Households or individuals? Adjusted for household size or not? Market or disposable? Wealth Ginis run far higher than income Ginis for the same country because assets are more concentrated than earnings, and where households hold negative net worth a wealth Gini can even exceed 1. Household surveys also miss top incomes, since the very rich respond less and under report more, which biases measured inequality downward. Two Ginis built on different definitions are two different statistics wearing the same name.

There is one more technical quirk worth carrying. The Gini responds more strongly to income transfers among the crowded middle of a distribution than to equally sized transfers out at the sparse extremes, because the weight it gives a transfer depends on how many households sit between the two parties in the ranking. A measure that reacts most to the middle is not the measure to reach for if the question you care about is what is happening at the very top.

Three alternatives worth knowing

The quintile ratio (S80/S20). Divide the top fifth's income share by the bottom fifth's. In Economy X that is 48 / 5 = 9.6, meaning the richest fifth collectively receive 9.6 times what the poorest fifth receive. After taxes and transfers it falls to 40 / 8 = 5.0. The appeal is that anyone can compute it and everyone can interpret it. The weakness is that it discards the middle 60 percent entirely, so any change in the three middle quintiles leaves it completely unmoved.

The Palma ratio. The share of the top 10 percent divided by the share of the bottom 40 percent. The reasoning behind it is that the middle and upper middle of the distribution tend to hold a fairly stable share of national income, so the interesting variation lives in the two tails. Quintile data alone cannot produce a Palma ratio, since the top decile is buried inside the top quintile, and you need decile data. If a decile breakdown of Economy X showed the top 10 percent taking 33 percent of income, the Palma would be 33 / 15 = 2.2, since the bottom 40 percent hold 5 + 10 = 15 percent. Higher values mean more inequality, and unlike the Gini the number has a direct verbal reading.

The Theil index. Borrowed from information theory, it compares each household's income with the mean and sums the deviations in logarithmic form, T = the average across households of (y / mean) x ln(y / mean). It is 0 under perfect equality and rises with inequality, with an upper bound of ln(N) rather than a tidy 1, which makes the raw value harder to interpret than a Gini. Its genuine advantage is that it is additively decomposable: total inequality splits cleanly into a within group component and a between group component. That lets you answer questions the Gini cannot, such as how much of a country's inequality comes from gaps between regions or between education levels rather than from spread inside those groups.

None of the three replaces the Gini. The standard practice is to report the Gini for comparability, plot the Lorenz curve for shape, and add a tail sensitive measure when the question is specifically about the top or the bottom.

Where this shows up in your course

The Lorenz curve and Gini coefficient travel further than most diagrams in economics, which is why it pays to learn them properly once.

• AP Microeconomics meets them in the treatment of income distribution and government redistribution, alongside market failure and the case for intervention.
• AP Macroeconomics does not list them in its course framework, but the distinction between growth that raises average income and growth that spreads it evenly is worth carrying over from micro when you write about living standards.
• IB Economics uses them wherever the course deals with equity in the distribution of income, and the IB economics guide sets out the surrounding material. Check your current syllabus for whether you are expected to calculate a Gini or only to interpret one.
• A-level Economics uses them in the inequality and redistribution material within macro policy, covered in the A-level economics guide.
• College introductory courses, both micro and macro, use the pair when introducing welfare economics and public finance.
• Sociology, geography, politics and development studies all use the same two tools, usually for interpretation rather than calculation, in units on stratification, regional development and welfare states.

Whatever the course, the marks come from the same three moves: describing the axes precisely, computing the coefficient correctly with the arithmetic visible, and then saying what the number does not tell you. Get comfortable with the definitions in the Gini coefficient and Lorenz curve entries, then work through the surrounding policy material in the market failure module or the wider learn economics hub.