A level Maths statistics

By The STEM-Tutor team · Updated October 2026

Statistics is part of every A level Maths course. Here are the key ideas, with a focus on distributions and hypothesis testing.

Sampling and data

• Sampling methods: simple random, systematic, stratified, quota, opportunity and cluster sampling.
• Measures of location and spread, and representing data with histograms, box plots and cumulative frequency.
• Correlation and regression, and interpreting them in context.

Check whether your board uses a large data set; questions may assume you know it.

Probability

Use Venn diagrams and tree diagrams. Understand mutually exclusive and independent events, and conditional probability.

The binomial distribution

X ~ B(n, p) models the number of successes in n independent trials with a fixed probability p. Check the conditions in context before using it.

The normal distribution

X ~ N(μ, σ²) is symmetric about its mean. Use your calculator for probabilities, and the standard normal Z = (X − μ)/σ for finding unknown means or standard deviations.

Hypothesis testing

1. State H₀ and H₁ in terms of the parameter.
2. State the significance level and whether the test is one-tailed or two-tailed.
3. Find the probability of the observed result or worse, or find the critical region.
4. Compare with the significance level and conclude in context, without claiming to prove anything.

Year 2 also covers tests for a normal mean and for correlation.

Common mistakes

• Halving the significance level in a two-tailed test, then forgetting to use it on both tails.
• Concluding "H₀ is true" instead of "insufficient evidence to reject H₀".
• Missing context in the final conclusion.

The big ideas in A level statistics

A level statistics asks you to model real situations with probability distributions and then test claims using data. The binomial distribution models counts of successes; the normal distribution models continuous measurements such as heights or times. Hypothesis testing ties these together: you assume a model, see how surprising your data would be, and decide whether there is evidence against it.

Worked examples

Example 1: a binomial probability

X ~ B(20, 0.3). Find P(X ≤ 4).

• Use your calculator's cumulative binomial function: P(X ≤ 4) ≈ 0.2375.

Example 2: a binomial hypothesis test

A coach claims 30% of shots are missed. After training, a player misses 2 shots out of 20. Test at the 5% level whether the miss rate has fallen.

• H₀: p = 0.3; H₁: p < 0.3. Under H₀, X ~ B(20, 0.3).
• P(X ≤ 2) ≈ 0.0355, which is less than 0.05.
• Reject H₀: there is evidence at the 5% level that the miss rate has fallen.

Example 3: a critical region

For the test above, find the critical region.

• P(X ≤ 2) ≈ 0.0355 and P(X ≤ 3) ≈ 0.1071.
• The largest region with probability under 5% is X ≤ 2; the actual significance level is about 3.55%.

Example 4: a normal probability

X ~ N(50, 4²). Find P(X > 56).

• z = (56 − 50)/4 = 1.5, so P(X > 56) = P(Z > 1.5) ≈ 0.0668.

Example 5: a correlation test

A sample of 10 gives PMCC r = −0.6. Test for negative correlation at the 5% level (critical value 0.5494).

• H₀: ρ = 0; H₁: ρ < 0.
• |−0.6| > 0.5494, so reject H₀: there is evidence of negative correlation.

Sampling methods at a glance

• Simple random: every sample of the chosen size is equally likely.
• Systematic: every kth item from an ordered list, after a random start.
• Stratified: random samples from each group, in proportion to its size.
• Quota: the interviewer fills quotas for each group, without randomness.
• Opportunity: whoever is available; quick but likely to be biased.

More mistakes to avoid

• Not stating hypotheses in terms of a population parameter such as p or μ.
• Forgetting to halve the significance level for a two-tailed test.
• Writing a conclusion with no context.
• Using the binomial distribution when trials are not independent or p is not constant.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Do I need to learn the large data set?

If your board uses one, yes. Questions may assume familiarity with its variables and context.

Can I use my calculator for distributions?

Yes. Make sure your calculator has binomial and normal functions and that you know how to use them quickly.

Key takeaways

• Use the binomial distribution for counts of successes in fixed, independent trials.
• Use the normal distribution for continuous data; standardise with z.
• State hypotheses about population parameters.
• Compare probabilities or test statistics with the significance level.
• Always conclude in context.

Where to go next

Review the rest of the course alongside statistics.

• A level Maths topics
• A level Maths formula booklet
• A level Maths mechanics

Interpreting results in context

Many statistics marks are for interpretation rather than calculation. A conclusion like "reject H₀" earns little on its own. Write what it means in the setting of the question, for example "there is evidence at the 5% level that the new training has reduced the miss rate". Use cautious language: a test gives evidence, not proof. If you do not reject H₀, say there is insufficient evidence, not that H₀ is true.

When you criticise a model, link your point to the context. Saying "the trials may not be independent" is better when you add why: shots taken when tired may be more likely to miss, so the probability may change through a session.

Data presentation and interpretation

Expect questions on histograms, box plots, cumulative frequency and scatter diagrams. With histograms, remember that area represents frequency, so frequency density equals frequency divided by class width. With box plots, an outlier is often defined as more than 1.5 times the interquartile range beyond a quartile, but always use the rule the question gives. Interpolation lets you estimate the median and quartiles from grouped data.

A note on the large data set

Boards that use a large data set expect you to know its context, such as what each variable measures, where the data was collected and any unusual values. You will not need to memorise numbers, but questions may ask you to comment on whether a sample from it is representative or to spot an impossible value. Spend an hour exploring it with a spreadsheet before the exam; that small investment often pays back several marks.

Statistics rewards clear thinking and careful wording more than heavy algebra, so it can become a strong section with steady practice.