Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

Measuring inequality: fitting Lorenz curves and computing the Gini coefficient

AA SLAA HLAI SLAI HL Solid Also in: Calculus, Statistics

Research question

Which function best fits the Lorenz curve of income in [two countries], and how close is the Gini coefficient calculated by integrating that function to the officially published value?

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Why it makes a good exploration

The Gini coefficient is quoted in the news but rarely explained. It is an area between two curves — a direct use of integration — and fitting the Lorenz curve is a genuine modelling decision.

The mathematics you'll need

Lorenz curves from income shares by quintile
Fitting a function such as L(x) = xᵏ or other forms
Definite integration: G = 1 − 2∫₀¹ L(x) dx
Trapezoidal rule comparison

Where the data comes from

Income shares by quintile or decile from the World Bank or Our World in Data.

World Bank Open Data — Development indicators (GDP, life expectancy, mortality, internet use) for every country, usually from 1960.

Real data for this idea

Primary source: World Bank Open Data: Poverty and Inequality Platform. The country profiles give income or consumption shares by decile: the points of a Lorenz curve. Licence: CC BY 4.0 for most datasets (the World Bank's licence page lists any exceptions).
Also useful: World Bank Open Data: Gini index (SI.POV.GINI). Many countries have gaps, so use each country's latest survey year and record which year it is.
Readable chart: Our World in Data: Gini coefficient. Use it to see the pattern; download your numbers from the primary source or the chart's own download. Cite as: Our World in Data, ‘Gini coefficient’, CC BY 4.0. Original source: World Bank Poverty and Inequality Platform (2026).

Some countries' Gini figures measure income and others consumption, and the two are not directly comparable: compare like with like. Links checked October 2026; we link to the data and never copy it. More free data sets.

A possible outline

Explain the Lorenz curve and the Gini coefficient.
Plot the cumulative shares for two countries.
Fit models and choose the best.
Integrate to find G; compare with the trapezoidal estimate and official figures.
Reflect on what the Gini hides.

Pitfalls that cost marks

Models that do not pass through (0,0) and (1,1).
Mixing income and wealth data.
Not explaining why the trapezoidal estimate differs.

Showing personal engagement

Compare your own country with another you know.
Discuss a policy change and its effect on the curve.
Explain Gini to a family member.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Lorenz curves from income shares by quintile; Fitting a function such as L(x) = xᵏ or other forms
AA HL	Good fit	Lorenz curves from income shares by quintile; Fitting a function such as L(x) = xᵏ or other forms
AI SL	Good fit	Lorenz curves from income shares by quintile; Fitting a function such as L(x) = xᵏ or other forms
AI HL	Good fit	Lorenz curves from income shares by quintile; Fitting a function such as L(x) = xᵏ or other forms

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: models that do not pass through (0,0) and (1,1); and mixing income and wealth data. Say how each one affects your answer.

Use of mathematics (E)

SL: lorenz curves from income shares by quintile; fitting a function such as L(x) = xᵏ or other forms; definite integration: G = 1 − 2∫₀¹ L(x) dx; trapezoidal rule comparison, each explained in your own words and carried out correctly.

How finance explorations reach the top bands in general: Finance & economics IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Compare the trend in Gini over 30 years and discuss whether your model's parameter is a better measure.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL finance exploration adds.

See a complete IA, marked

Should I lease or buy my first car? (AI SL) is a complete finance exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Lorenz curves from income shares by quintile, fitting a function such as L(x) = xᵏ or other forms, definite integration: G = 1 − 2∫₀¹ L(x) dx and trapezoidal rule comparison.

Where can I get data for this IA?

Income shares by quintile or decile from the World Bank or Our World in Data. Checked sources and tools: World Bank Open Data and Our World in Data.

Turn this idea into your IA

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