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IB Math AI SL · Topic 4

Statistics and probability

The largest topic in the course — eleven sections — and the one where the exam rewards language as much as calculation. Computing a p-value is routine; concluding from it in context is where grades separate.

What the syllabus covers

Eleven assessed sections, SL 4.1 to 4.11 — from sampling bias through to formal hypothesis tests.

SL 4.1SamplingPopulation versus sample, random sampling techniques and their effectiveness, bias and reliability of sources, and the interpretation of outliers.
SL 4.2Presentation of dataFrequency tables, histograms, box-and-whisker diagrams, and cumulative frequency graphs from which medians, quartiles and percentiles are read.
SL 4.3Central tendency and dispersionMean, median and mode; estimating the mean from grouped data; IQR and standard deviation, including how constant changes to the data move them.
SL 4.4Correlation and regressionScatter diagrams, lines of best fit by eye through the mean point, Pearson's r, and the regression line y = ax + b with its parameters interpreted.
SL 4.5Probability conceptsTrials and equally likely outcomes; P(A) = n(A)/n(U); complementary events; expected number of occurrences.
SL 4.6Combined eventsVenn, tree and sample-space diagrams; the addition rule; mutually exclusive, conditional and independent events.
SL 4.7Discrete random variablesProbability distributions and the expected value E(X) = Σxᵢ·P(X = xᵢ), applied to games and insurance-style contexts.
SL 4.8Binomial distributionNotation X ~ B(n, p) with mean np and variance np(1 − p); individual probabilities via the GDC.
SL 4.9Normal distributionProbabilities and inverse-normal calculations on the GDC, plus judging whether the model is reasonable for the data.
SL 4.10Spearman's rankWhen rank correlation is the better measure than Pearson's r, and how outliers affect each differently.
SL 4.11Hypothesis testingFormulating H₀ and H₁, significance levels and p-values; χ² tests for independence and goodness of fit; the t-test; one- and two-tailed decisions.

How it is examined

Hypothesis testing appears in SL Paper 2 exactly as it does beyond school statistics — as a procedure whose steps must all be visible.

  • Recent SL papers embed χ² tests and t-tests inside real scenarios — spending by weekend customers, crop yields — and award separate marks for hypotheses, p-value, and conclusion.

  • Conclusions must compare the p-value to the significance level and interpret the decision back into the scenario's language; 'reject H₀' alone is incomplete.

  • Regression questions check direction: predicting y requires the y-on-x line, a distinction examiners regularly penalise.

  • Distribution work (binomial, normal, inverse normal) is GDC-based but expects the setup — distribution chosen and parameters stated — before the output.

  • The booklet devotes five pages here — probability rules, expected value, binomial mean and variance — but no procedures at all: there is no normal formula and no χ² recipe to copy. The tests live in the GDC's statistics menu, so choosing which test to open is itself the examined skill.

  • Real papers also test sampling judgement in words — naming whether a plan was convenience, systematic or stratified sampling, then stating one disadvantage of it.

A marked question, mark by mark

A normal-distribution exercise with the contextual sentence that decides the final mark.

Rehearse the whole testing procedure, marked step by step

From SL 4.1 sampling to SL 4.11 t-tests, every generated exercise comes back with each mark awarded or withheld and a reason — including the contextual sentences students habitually omit.

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New to the codes? M1, A1, R1 explained →

Normal distribution · SL 4.9

Seed 2E87

Battery lifetimes are modelled by a normal distribution with mean 120 hours and standard deviation 8 hours.

  1. (a)Find the probability that a battery lasts fewer than 110 hours.
  2. (b)Interpret this probability in context.

Your work

z = (110 − 120)/8 = −1.25

P(Z < −1.25) ≈ 0.106

Marks awarded

  • M1Mark awarded. Standardised correctly with μ = 120 and σ = 8 before applying technology.
  • A1Mark awarded. 0.106, correct to three significant figures.
  • R1Mark not awarded. No interpretation: the 0.106 was never linked to the proportion of batteries failing before 110 hours.

Where marks are actually lost

In this topic the calculator is rarely the problem — the surrounding language is.

Skipping the formal statement of H₀ and H₁ at the start of a test.

CostThe first marks of every hypothesis question are given away before any calculation happens.

Ending with 'reject H₀' instead of comparing the p-value to the significance level in a sentence about the scenario.

CostThe conclusion mark is explicitly reserved for the comparison and its meaning.

Predicting with the x-on-x line, or reading r ≈ ±1 as causation.

CostDirection of prediction and correlation-versus-causation are standing examiner penalties.

Trusting Pearson's r on clearly monotonic-but-nonlinear data where Spearman's rank is appropriate.

CostAppropriateness-of-measure judgements are marked, especially in the presence of outliers.