IA idea · Matrices, transformations & Markov chains

Ranking teams that haven't all played each other: the Colley matrix

AI HLAA HL Ambitious Also in: Sport, Statistics

Research question

How does a ranking from solving the Colley linear system compare with a league table halfway through a season, and which is better at predicting second-half results?

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

Ranking from incomplete schedules is a real problem (college sports, chess). The method is a single matrix equation, and testing its predictions against real results gives a clear answer.

The mathematics you'll need

Setting up the Colley system Cr = b (new: explain it)
Solving with matrices
Comparing rankings with Spearman's rank correlation
Prediction accuracy as a proportion
Effect of strength of schedule

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.

The statistics, step by step

Worked with every number shown, with what examiners look for and the common mistakes: Spearman's rank correlation · Pearson's correlation and regression. Then run the same steps on your own data in Analyse my data, or start from the statistics workflow.

Where the data comes from

Use one season of results from Football-Data.co.uk (or another sport you follow).

Football-Data.co.uk — Match results, shots, cards and bookmaker odds for 25+ seasons of European leagues, CSV.

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 method with a four-team example.
Build the system for the first half of a season.
Compare with the league table.
Test both on second-half results.
Reflect on draws, home advantage and margin of victory.

Pitfalls that cost marks

Using a ranking method you can't explain.
Comparing rankings only by eye.
Ignoring draws without comment.

Showing personal engagement

Use your own league.
Rank your school's sports teams.
Invent an adjustment for home advantage.

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	Good fit	Setting up the Colley system Cr = b (new: explain it); Solving with matrices
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	Setting up the Colley system Cr = b (new: explain it); Solving with matrices

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: using a ranking method you can't explain — 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

Compare with a Massey (points difference) ranking or an Elo rating.

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).