Normal Distribution in 9709 S1: The Step-by-Step Method Examiners Reward

The normal distribution appears in almost every Cambridge 9709 Probability & Statistics 1 paper, and it is one of the most methodical topics in the course. The same four steps solve nearly every question. Marks are lost not through hard mathematics, but through skipped working, wrong tails and careless table reading.

The syllabus says it directly: "For calculations involving standardisation, full details of the working should be shown." This guide shows you what "full details" means in practice.

Where it sits

• Component: Paper 5, Probability & Statistics 1 (S1), syllabus section 5.5.
• Paper 5: 1 hour 15 minutes, 50 marks, 6–8 structured questions; 40% of the AS Level and 20% of the A Level.
• Compulsory for A Level: every 9709 A Level route includes Paper 5.
• Tables: normal distribution tables are in the formula list (MF19).

What the syllabus requires

1. Use a normal distribution to model a continuous random variable, and use normal tables.
2. For X ~ N(μ, σ²), find P(X > x₁) or related probabilities given μ and σ.
3. Work backwards: find a relationship between x₁, μ and σ given a probability.
4. Use the normal approximation to the binomial, with a continuity correction, when np > 5 and nq > 5.

Note that the notation is N(μ, σ²). The second parameter is the variance, not the standard deviation. Misreading N(50, 16) as σ = 16 (instead of σ = 4) is one of the most common errors.

The four-step method

Step 1: Define and sketch. Write X ~ N(μ, σ²) and draw a quick bell curve. Mark the mean and shade the region you need. The sketch prevents most tail errors.

Step 2: Standardise. Convert to Z ~ N(0, 1) with Z = (X − μ)/σ. Write this line out in full every time; it is where the method mark sits.

Step 3: Use the table correctly. Tables give Φ(z) = P(Z < z) for z ≥ 0. Use symmetry: - P(Z > z) = 1 − Φ(z) - P(Z < −z) = 1 − Φ(z) - P(a < Z < b) = Φ(b) − Φ(a)

Step 4: Answer in context, to 3 significant figures unless told otherwise.

Worked example 1: a forward problem (original question)

The masses of apples from an orchard are normally distributed with mean 150 g and standard deviation 12 g. Find the probability that a randomly chosen apple has mass between 135 g and 168 g.

X ~ N(150, 12²)

P(135 < X < 168) = P((135 − 150)/12 < Z < (168 − 150)/12) = P(−1.25 < Z < 1.5)

= Φ(1.5) − (1 − Φ(1.25)) = 0.9332 − (1 − 0.8944) = 0.9332 − 0.1056 = 0.828 (3 s.f.)

The examiner wants to see the standardised values −1.25 and 1.5 written down, and the correct combination of table values.

Worked example 2: an inverse problem (original question)

The time taken to complete a puzzle is normally distributed with mean μ minutes and standard deviation 4 minutes. 10% of people take longer than 30 minutes. Find μ.

P(X > 30) = 0.10, so P(Z > z) = 0.10, which gives Φ(z) = 0.90 and z = 1.282 (from the critical-values table).

(30 − μ)/4 = 1.282 → μ = 30 − 5.128 = 24.9 minutes (3 s.f.)

The key check: 30 is in the upper 10%, so it must be above the mean, and z must be positive. If your z is negative here, your sketch will show the mistake.

Worked example 3: two unknowns (original question)

X ~ N(μ, σ²). P(X < 40) = 0.2 and P(X > 70) = 0.1. Find μ and σ.

• P(X < 40) = 0.2 → z = −0.842, so (40 − μ)/σ = −0.842 → 40 = μ − 0.842σ
• P(X > 70) = 0.1 → z = 1.282, so (70 − μ)/σ = 1.282 → 70 = μ + 1.282σ

Subtracting: 30 = 2.124σ → σ = 14.1, then μ = 40 + 0.842 × 14.12 = 51.9 (3 s.f.)

Keep extra figures in σ before substituting back, or the final answer drifts.

Normal approximation to the binomial

When X ~ B(n, p) with large n, use Y ~ N(np, npq), provided np > 5 and nq > 5. State this check in your answer; it is often worth a mark.

Continuity correction: a discrete value k covers the interval k − 0.5 to k + 0.5. | Binomial | Normal | |---|---| | P(X ≤ 20) | P(Y < 20.5) | | P(X < 20) | P(Y < 19.5) | | P(X ≥ 20) | P(Y > 19.5) | | P(X = 20) | P(19.5 < Y < 20.5) |

Original example: a fair coin is thrown 100 times. Estimate P(at least 60 heads). np = nq = 50 > 5. Y ~ N(50, 25), σ = 5. P(X ≥ 60) ≈ P(Y > 59.5) = P(Z > 1.9) = 1 − 0.9713 = 0.0287

Reading the question: the language of S1

Many lost marks start with misreading. Translate the wording before you calculate: - "At least 30" means X ≥ 30; "more than 30" means X > 30. For a continuous variable these give the same probability, but for the binomial they don't. - "Find the value of k such that 85% of values lie below k" is an inverse problem: Φ(z) = 0.85. - "Symmetrical about the mean" questions often need only symmetry, with no table at all. For example, if P(X > μ + a) = 0.2, then P(μ − a < X < μ + a) = 1 − 2(0.2) = 0.6. - "Expected number" means multiply the probability by the sample size, e.g. 0.828 × 200 apples ≈ 166 apples.

Read twice, sketch once, then calculate.

Six mistakes that cost marks

1. Variance vs standard deviation: N(μ, σ²) gives the variance.
2. No standardisation line: jumping straight to a probability loses the method mark.
3. Wrong tail: forgetting 1 − Φ(z) for "greater than".
4. Sign of z in inverse problems: below-mean values have negative z.
5. Missing or wrong continuity correction, or using one when the variable is already continuous.
6. Over-rounding z before the final step.

How to practise

• Start with 9709 S1 normal distribution topical questions from Kambridge Topicals (Paper 5 filter), mark schemes interleaved.
• Then sit full Paper 5s from the 9709 past papers library under a 75-minute limit.
• Revisit the 9709 Paper 1 guide if algebra is slowing your inverse problems.

Frequently asked questions

Do I need to use the critical-values table, or can I use a calculator? The tables in MF19 are standard, and some calculators have normal functions. Either way, show the standardisation and the z-value clearly so method marks can be awarded.

When do I use a continuity correction? Only when approximating a discrete distribution (the binomial) with the continuous normal distribution.

How many marks is normal distribution worth? It varies by paper. It typically appears as one substantial question, sometimes combined with the binomial approximation.

Get S1 right first time

Dr. Rizwan Jahangir teaches S1 methodically: sketch, standardise, read, interpret. Once that routine is automatic, normal distribution becomes some of the most dependable marks on Paper 5.

Sources: Cambridge International AS & A Level Mathematics 9709 syllabus for examination in 2026–2027: section 5.5, the Paper 5 assessment overview, the routes table and MF19. https://www.cambridgeinternational.org/Images/697427-2026-2027-syllabus.pdf. Table values (Φ(1.25) = 0.8944, Φ(1.5) = 0.9332, Φ(1.9) = 0.9713; z = 1.282 for 0.9 and 0.842 for 0.8) are standard normal values.

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