Mathematics 9709 · syllabus 5.5
Cambridge A Level Mathematics 9709 — The normal distribution topical past papers
Standardising N(μ, σ²) to find probabilities and unknown means or standard deviations, and approximating a binomial with a continuity correction.
- Questions
- 156
- Years
- 2009–2026
- With mark scheme
- 100%
- Questions 2022–2026
- 41
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What The normal distribution covers
The normal distribution models continuous quantities that cluster symmetrically around a mean, such as lengths or masses. In section 5.5 of Paper 5 you write X ~ N(μ, σ²), standardise with Z = (X − μ)/σ, and use tables of Φ(z) = P(Z ≤ z) to find probabilities, including probabilities between two values. A sketch of the curve with the area you want shaded keeps the reasoning straight.
Questions also run backwards: given a probability, you find the z-value from the tables and then an unknown value of x, μ or σ, and two such conditions give simultaneous equations for μ and σ. The last skill is approximating B(n, p) by N(np, np(1 − p)) when n is large enough that np > 5 and n(1 − p) > 5, using a continuity correction because a discrete count is being modelled by a continuous distribution.
The section brings together earlier Paper 5 work: the mean and standard deviation from representation of data, and the binomial distribution from section 5.4, which the normal approximation stands in for when adding up exact binomial probabilities would take too long. The normal distribution is then the foundation of most of Paper 6: linear combinations of normal variables, the distribution of a sample mean, confidence intervals and hypothesis tests about a mean all rely on it.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 5.5 The normal distribution
What you should be able to do
- Standardise a normal variable and use tables to find probabilities
- Find probabilities between two values, using the symmetry of the curve
- Use the tables in reverse to find an unknown value, mean or standard deviation
- Solve simultaneous equations for μ and σ from two given probabilities
- Approximate a binomial distribution by a normal one, with a continuity correction
- Sketch normal curves to show the probability being found
Key ideas
- If X ~ N(μ, σ²), then Z = (X − μ)/σ ~ N(0, 1)
- By symmetry, P(Z ≤ −z) = 1 − Φ(z)
- Useful values: Φ(1) = 0.8413, Φ(1.645) = 0.95, Φ(1.96) = 0.975, Φ(2) = 0.9772
- B(n, p) ≈ N(np, np(1 − p)) when np > 5 and n(1 − p) > 5
- Continuity correction: a binomial P(X ≤ 12) becomes P(Y < 12.5) for the approximating normal variable Y
How The normal distribution has been examined
From 2022 to 2026, The normal distribution was on Paper 5 · Probability & Statistics 1 in 14 of 14 exam series, worth about 21% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 5 · Probability & Statistics 1 | 41 | 21% | 14 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 9.
Filed in Momentum under: The Normal Distribution.
A worked example
The masses of apples from an orchard are normally distributed with mean 150 g and standard deviation 20 g. Find the probability that a randomly chosen apple has a mass between 130 g and 190 g, and find the mass exceeded by 5% of the apples.
- Let X ~ N(150, 20²). Standardise: (130 − 150)/20 = −1 and (190 − 150)/20 = 2.
- P(130 < X < 190) = Φ(2) − Φ(−1) = Φ(2) − (1 − Φ(1)).
- = 0.9772 − (1 − 0.8413) = 0.9772 − 0.1587 = 0.8185.
- For the heaviest 5%, P(X > m) = 0.05, so (m − 150)/20 = 1.645.
- m = 150 + 1.645 × 20 = 182.9.
Answer: P(130 < X < 190) = 0.819 (3 s.f.); 5% of the apples are heavier than 183 g
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Dividing by the variance instead of the standard deviation when standardising: if X ~ N(50, 16), then σ = 4, not 16.
- Losing the minus sign of z for a value below the mean, which gives a probability above 0.5 where it should be below.
- Leaving out the continuity correction, or applying it the wrong way: a binomial P(X ≥ 30) becomes P(Y > 29.5), not P(Y > 30.5).
- Using the normal approximation without checking that np and n(1 − p) both exceed 5.
Exam technique
- Show the standardisation in full, writing (x − μ)/σ with the numbers in, because the method marks depend on it.
- Sketch the curve and shade the region before using the tables; it shows whether you need Φ(z) or 1 − Φ(z).
- Give probabilities to 3 significant figures, and use z-values to at least 3 decimal places where the tables allow.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.