A-Level Maths statistics revision: distributions, tests, traps
A-Level statistics covers sampling methods and bias, presenting and interpreting data, probability (Venn, trees, conditional), the binomial and normal distributions including inverse-normal problems, and hypothesis testing with significance levels and critical regions. The questions repeat almost exactly year to year — learn the hypothesis-test skeleton, use your calculator's distribution functions fluently, and always conclude in context, and statistics becomes the most reliable third of the applied paper.
Statistics is where well-prepared students bank their most reliable marks and where under-prepared ones leak them, and the difference is rarely intelligence — it is knowing that the questions repeat. A hypothesis test in 2025 is structurally the same question as in 2019 with different numbers. Learn the skeletons and this becomes the easiest third of the applied paper.
The specification, in plain English
| Area | What's in it | The trap |
|---|---|---|
| Sampling | Random, systematic, stratified, quota, opportunity — and their biases | The 'give a criticism' line worth a mark |
| Data presentation | Histograms, box plots, cumulative frequency, outliers | Histogram y-axis is frequency density, not frequency |
| Probability | Venn diagrams, trees, conditional probability, independence | P(A|B) is not P(B|A) |
| Binomial | X ~ B(n, p), conditions, cumulative probabilities | Using P(X = k) where the tail is needed |
| Normal | X ~ N(μ, σ²), standardising, inverse normal | Reading the wrong tail in inverse problems |
| Hypothesis tests | One/two-tailed, significance level, critical regions, p-values | Conclusion must be in context — every time |
The binomial's four conditions
Examiners regularly ask why a binomial model is appropriate, and the mark scheme wants the conditions stated: a fixed number of trials, each with two outcomes, a constant probability of success, and independence between trials. Learn the four as a set — 'fixed n, two outcomes, constant p, independent' — because the question is worth a mark or two nearly every series.
The hypothesis-test skeleton
- State H₀ and H₁ in terms of the parameter — H₀: p = value, H₁: p > value (or <, or ≠ for two-tailed).
- Name the distribution under H₀: X ~ B(n, p) or N(μ, σ²).
- Calculate the tail probability in the direction H₁ points — never the point probability.
- Compare with the significance level (halved each tail for two-tailed).
- Conclude in context: 'there is / is not significant evidence that…' with the scenario named.
Worked questions
A manufacturer claims 5% of components are faulty. In a batch of 50, 6 are faulty. Test at the 5% level whether the true rate exceeds 5%.
- H₀: p = 0.05, H₁: p > 0.05. Under H₀, X ~ B(50, 0.05).
- We need P(X ≥ 6) — the upper tail, because H₁ says 'exceeds'.
- P(X ≥ 6) = 1 − P(X ≤ 5) = 1 − 0.9622 ≈ 0.0378.
- 0.0378 < 0.05, so the result is significant.
- In context: there is evidence at the 5% level that the faulty rate exceeds 5%.
Reject H₀ — P(X ≥ 6) ≈ 0.038 < 0.05. Note how the tail probability, not P(X = 6), is the whole test.
Masses are modelled as X ~ N(500, 15²) grams. Find the mass exceeded by only 5% of items.
- We want x with P(X > x) = 0.05 — equivalently P(X < x) = 0.95.
- Inverse normal on the calculator: μ = 500, σ = 15, area 0.95 gives x ≈ 524.7.
- Alternatively standardise: z = 1.6449, x = 500 + 1.6449 × 15 ≈ 524.7.
- Check the direction — 'exceeded by only 5%' is the upper tail, so x is above the mean. If your answer is below 500, you read the wrong tail.
x ≈ 524.7 g. Inverse-normal questions are tail-reading exercises — sketch the curve and shade the right side before computing.
P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.15. Find P(A|B) and state whether A and B are independent.
- P(A|B) = P(A ∩ B)/P(B) = 0.15/0.5 = 0.3.
- Independence test: P(A|B) should equal P(A) if independent — 0.3 ≠ 0.4.
- Alternatively check P(A) × P(B) = 0.2 ≠ 0.15 = P(A ∩ B).
- State the conclusion explicitly — 'not independent' is the answer, not just the numbers.
P(A|B) = 0.3; not independent, since P(A|B) ≠ P(A).
Where marks get dropped
- P(X = k) instead of the tail — the single commonest error in statistics papers.
- Forgetting to halve the significance level in a two-tailed test.
- Concluding 'accept H₀' or 'reject H₀' with no context sentence — the final mark is for the real-world statement.
- Using a sample statistic as the parameter in H₀ — hypotheses are about the population parameter.
- In histograms, reading frequency off the y-axis directly — it's frequency density, so area is what counts.
- Stating 'correlation implies causation' reasoning — the criticism mark wants you to reject exactly that.
Your statistics checklist
- I can recite the four binomial conditions unprompted.
- I can run a hypothesis test start-to-finish: hypotheses, distribution, tail, compare, context.
- I can find cumulative binomial and normal probabilities on my calculator in seconds.
- I can do inverse-normal problems and know which tail I'm reading.
- I can name the bias in each sampling method and criticise a sample design.
- I write 'in context' conclusions as full sentences about the scenario.
- Edexcel students: I have explored the large data set and know its variables.
Statistics answers are marked partly on phrasing — which is awkward, because no mark scheme teaches you the phrasing, it just withholds the mark. If your conclusions keep losing the last mark, show them to Lumi; it will tell you which word the examiner was waiting for and drill you until the skeleton is reflex.
What to take from this
- Statistics is the most repeatable part of the course — the question skeletons barely change year to year.
- A hypothesis test is: hypotheses, distribution under H₀, tail probability, comparison, conclusion in context.
- Use your calculator's binomial, normal and inverse-normal functions — doing tables by hand is a time leak.
- 'In context' is not decoration: the final mark of every test question is for a sentence about the real world.
- Correlation is not causation — the criticism mark on data questions almost always wants this or sampling bias.
Questions people also ask
Both test the same thing. The critical region is the set of outcomes extreme enough to reject H₀ — you check whether your observation falls inside it. The p-value is the probability of a result at least as extreme as yours under H₀ — you compare it to the significance level. Exams accept either; p-values are faster on the calculator.
Only for two-tailed tests — where H₁ says p ≠ value rather than p > or p <. Each tail gets half the significance level, so a 5% two-tailed test uses 2.5% per tail.
Yes — the final mark of essentially every hypothesis-test question is reserved for it. 'Reject H₀' is a statement about symbols; the examiner wants 'there is insufficient evidence that the coin is biased'.
Every allowed calculator has a Normal CD function for cumulative probabilities and an Inverse Normal for the reverse problems. Learn both cold — doing these via tables and standardisation is accurate but slow, and statistics sections are time-tight.