IA idea · Matrices, transformations & Markov chains

Projecting a country's age structure with a Leslie matrix

AI HL Ambitious Also in: Environment, Modelling

Research question

Using UN birth and survival rates by age group, what age structure does a Leslie matrix predict for a country in 50 years, what long-run growth rate does its dominant eigenvalue give, and how do the projections compare with the UN's own?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

Leslie matrices are a real demographic tool. Real UN data, a clear long-run answer from eigenvalues and a published projection to compare with make a complete investigation.

The mathematics you'll need

Building a Leslie matrix from fertility and survival rates
Matrix powers for projections
Eigenvalues and eigenvectors for long-run growth and age structure
Comparing projections with published ones
Sensitivity to fertility changes

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Download fertility and survival (or mortality) by age group from UN World Population Prospects for a country you choose; state the version and variant.

UN World Population Prospects 2024 — Official UN population estimates 1950–present and projections to 2100, CSV bulk download.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explain the Leslie model with a three-age-group example.
Build the matrix for your country from UN data.
Project forward and compare with the UN projection.
Find the dominant eigenvalue and stable age structure.
Reflect on migration and changing rates.

Pitfalls that cost marks

Mixing age-group widths with time steps.
Ignoring migration without comment.
Using eigenvalues from software without interpreting them.

Showing personal engagement

Choose a country you have a connection to.
Compare two countries with very different structures.
Find the fertility rate that gives zero long-run growth.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Building a Leslie matrix from fertility and survival rates; Matrix powers for projections

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Collect the data for your matrix yourself (counting transitions, measuring a shape), and choose the states or the transformation from a situation you care about.

Reflection (D)

Question the model's assumptions: is the process memoryless, are the probabilities constant, does the transformation preserve what it should? Say how each affects your conclusion. For this idea, start with: mixing age-group widths with time steps — say how it affects your answer.

Use of mathematics (E)

SL: Matrices are in AI HL. At SL, keep to small matrices you explain carefully as new mathematics, with every multiplication shown once and the result interpreted.

HL: Transition matrices, powers, steady states and eigenvalues used correctly, with diagonalisation or a general result derived, and the long-run behaviour interpreted in context.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Add migration as a vector term, or compare female-only and two-sex models.

Extending it for HL

Diagonalise the matrix to find a formula for the nth state, or compare the steady state with what your data actually shows.

See a complete IA, marked

Our annotated exemplar How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) asks a different question, but shows how a complete matrices & markov exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).