IA idea · Matrices, transformations & Markov chains

Does rain today predict rain tomorrow? A Markov chain for the weather

AI HLAA HL Solid Also in: Environment, Probability

Research question

Using a daily rainfall record for your town, how well does a two-state Markov chain (wet/dry) predict the lengths of wet and dry spells, and does adding a memory of two days improve it?

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

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Why it makes a good exploration

Weather is the classic Markov example, but testing it on real local data is rarely done. The prediction of spell lengths gives a sharp test of the Markov assumption.

The mathematics you'll need

Transition probabilities estimated from counts
Matrix powers and the steady state
Spell lengths as a geometric distribution
Chi-squared goodness of fit for spell lengths
Extending to a four-state chain for two-day memory

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.

Modelling the data, step by step

This idea compares models fitted to data. See it worked step by step, with a criterion tip at every step: Choosing and comparing models.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

The statistics, step by step

Worked with every number shown, with what examiners look for and the common mistakes: Chi-squared goodness of fit. Then run the same steps on your own data in Analyse my data, or start from the statistics workflow.

Where the data comes from

Find a daily rainfall record for a station near you; many national weather services publish daily station data free. Cite the station and years. A school weather station also works if it has a long record.

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

Clean the data and define 'wet'.
Estimate the transition matrix and steady state.
Predict the distribution of spell lengths and compare with the data.
Build the two-day-memory chain and compare.
Reflect on seasons and on the threshold for 'wet'.

Pitfalls that cost marks

Mixing seasons so the probabilities aren't constant.
Comparing only the steady state, which almost any model gets right.
No test of the spell-length prediction.

Showing personal engagement

Use your own town's record.
Try different wet thresholds and see what changes.
Compare summer with winter.

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	Transition probabilities estimated from counts; Matrix powers and the steady state
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	Transition probabilities estimated from counts; Matrix powers and the steady state

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

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 seasons so the probabilities aren't constant — 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

Fit separate chains for each season, or compare two climates.

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).