Maths IA · Calculus · topic 20 of 97
Measuring inequality: Lorenz curves and the Gini coefficient · IB Maths IA idea: guide, sample graph and step-by-step plan
a picture, then one numberTwo countries can have the same average income and very different lives. The Gini coefficient squeezes that gap into one number. You're going to see where the number comes from, and what it hides.
- ✓ you care how wealth and income are shared out
- ✓ you want an integration topic that uses real published numbers
- ✓ you like turning a picture into one number and then questioning that number
The big idea
Line everyone up from poorest to richest and plot the share of income held by the poorest p of the people. That is the Lorenz curve L(p). If income were equal, the curve would be the straight line L = p. The more the curve sags below it, the more unequal the country. The Gini coefficient is twice the area between the line and the curve, so it is 1 − 2 times an integral of L. Real data give only a few points, one per quintile or decile, so you fit a curve through them and compare two ways of finding the area: the trapezium rule on the raw points, and integration of the fitted curve. The marks come from the comparison: how much the two differ, and what different curves, countries or years can share the same Gini.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Fit a one-parameter family such as L = p^k to quintile or decile shares from a statistics office or the World Bank.
→ A model chosen and tested. Linearise with logarithms to fit k, and measure how well the curve matches each data point.Find the Gini of the fitted curve by integration, as a formula in k.
→ A definite integral of a power function. For L = p^k you get a short formula in k, and it lets you invert: what k gives a Gini of 0.4?Find the Gini from the data points only, using trapeziums, and compare it with the integral.
→ Numerical against exact. For a curve that bends one way the trapezium rule always errs the same way; say which and why.Find two different Lorenz curves with the same Gini coefficient.
→ A counterexample that shows the limit of one number. Say what the two curves tell you that the Gini does not.Fit curves for two countries, or one country in two years, and compare their Ginis.
→ A statistical question: is the difference big enough to matter given the data are survey estimates?Fit L = 1 − (1 − p)^a as well and see which family matches the data better.
→ Two models compared, by residuals or a simple error measure. Explain which one you would use and why.Your Maths IA research question: run the check
which data? which fit?Write your own question first. Then make sure it ticks every box:
- Names the countries or years, the income measure (before or after tax, per person or household) and the source.
- Says how many data points you have (quintiles or deciles) and what you fit to them.
- Says what you compare: the Gini from integrating a fitted curve against the Gini from the trapezium rule.
- Says how you will judge the fit: for example the largest gap between the curve and a data point.
- Is honest about what one number hides, and about the fact that the data come from surveys.
The model
The Lorenz curve sags below the line of equality. The Gini coefficient compares the area between them with the whole triangle. Draw yours from your own data.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Why measure inequality, where your income data come from, and what you expect the Gini values to be.
How cumulative shares are built from the data, with a table and a first plot of the points.
The definition from areas, the link to 1 − 2 times the integral of L, and the extreme cases.
A family of curves, the fit by logarithms or by technology, and a check of the fit against the data.
The Gini of the fitted curve by integration, then by the trapezium rule, with the error explained.
Two countries or years side by side, with the difference and how sure you can be of it.
Back to your aim: the Ginis you found, how the two methods compare and what the number hides.
Survey data, few points, one number for a whole distribution. Say which limitation matters most.
Where every data table and definition came from.
What your Maths IA graph could look like
make the graph squareMade-up quintile data to show the form of a strong graph, not the answer: a curve fitted through shares. Yours will look different. example only
- The curve and the line of equality share axes from 0 to 1. A square graph makes the area easy to see.
- Axis labels say what is cumulative: people on x, income on y.
- Plot your data as points and draw the fitted curve through them, so the fit can be judged by eye.
- State the fitted parameter on the graph, with its Gini.
- Another graph to consider: two countries' Lorenz curves on one set of axes.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
What does the bend of the curve do?
For the family L = p^k, rank four values of k by their Gini. Work out the formula for yourself, and check that k = 1 gives complete equality.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out what a Lorenz curve is. Before any calculation, sketch the curve you expect for a country you know and guess its Gini.
- Choose the dataPick two countries or years and one income measure. Record the income share of each quintile or decile and the source.
- Build the curveTurn shares into cumulative shares, plot the points and check the last point is (1, 1).
- Trial run (1 lesson)Use made-up shares for L = p² and compute the Gini by hand. Check it is 1/3, then check the same value comes from your spreadsheet.
- Fit a curveFit L = p^k with logarithms or technology. Measure the gap between curve and points.
- IntegrateFind the Gini of your fitted curve as an integral and compare it with the trapezium rule on the raw points.
- CompareDo the same for the second country or year and say whether the difference matters.
- Go further, with a reasonFit a second family, or find two curves with the same Gini. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means in the real setting and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Power functions and logarithms – Linearise with and fit by regression through the origin or by technology.
- Definite integration and area – Compute and the Gini for the fitted curve.
- Trapezium rule – Find the area under the polyline through the data points and compare it with the exact area; relate the error to the curve's concavity.
- Model fit and error – Measure the gap between curve and data at each quintile and report the largest.
- Inverse of a function – Solve for to find the curve that has a chosen Gini.
Key equations
Syllabus links
- 2.9Exponential and logarithmic functions
- 5.3Differentiating powers of x
- 5.5Integration as anti-differentiation; areas
- 5.11Definite integrals and areas between curves
- 2.5Linear, quadratic, exponential, cubic and sinusoidal models
- 2.10Log scales, linearising data, log-log and semi-log graphsHL
- 4.4Correlation and linear regression
- 5.5Integration as anti-differentiation; areas
- 5.8Trapezoidal rule
Read up first
Data and sources to start from
Take income shares by quintile or decile for two countries or years from a statistics office or the World Bank, and record the survey year and income definition.
References to cite
- Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association, 9(70), 209-219.
- Atkinson, A. B. (1970). On the measurement of inequality. Journal of Economic Theory, 2(3), 244-263.
- World Bank. (n.d.). Poverty and Inequality Platform. https://pip.worldbank.org/
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
cumulative, poorest first- Plotting income shares for each group instead of cumulative shares.
- Sorting the groups from richest to poorest, which flips the curve above the line.
- Mixing income and wealth data, or before-tax and after-tax data, between countries.
- Treating a one-parameter curve as exact. Show how far it sits from each data point.
- Forgetting that the trapezium rule on a curve that bends one way always errs in one direction.
- Comparing two Ginis without saying that two quite different curves can share the same value.
More Maths IA ideas like this
Area under a curve: trapezium or Simpson?●○○
Simpson's paradox: when the overall trend lies●●○
Two shares or one? Diversification and portfolio risk●●○
Benford's law: does your data follow it?●●○
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.