A Level Maths 9709 · Paper 5 Probability & Statistics 1 · 5.5 · ~4 min read
The normal distribution
The shape and properties of the normal distribution, standardising with z = (x − μ)/σ, using tables forwards and backwards, finding unknown μ and σ, and the normal approximation to the binomial with a continuity correction.
Normal curveStandardisingΦ(z) tablesInverse normalFinding μ and σSymmetryNormal approximation to binomialContinuity correction
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Paper tag = where it is examinedFORMULA LIST (MF19) given in the examM1A1B1 how marks are awardedhighlight = comes up again and againTest yourself at the end
In 30 seconds
- X ~ N(μ, σ²): symmetric, bell-shaped, with about 68% of values within 1σ of the mean and 95% within 2σ.
- Standardise: z = (x − μ)/σ turns any normal into Z ~ N(0, 1), whose cumulative probabilities Φ(z) are in the tables.
- Working backwards: from a probability, find z in the tables, then solve (x − μ)/σ = z.
- Approximating B(n, p): use N(np, np(1 − p)) when np > 5 and n(1 − p) > 5, with a continuity correction of 0.5.
By the end you can
- Use the normal distribution and its symmetry, with tables of Φ(z).
- Find probabilities, and values of x from given probabilities.
- Find unknown μ and/or σ from given probabilities, using simultaneous equations.
- Use the normal approximation to the binomial, with a continuity correction, and know when it is suitable.
1Probabilities
Standardising
Z = (X − μ)/σ ; P(X < x) = Φ[(x − μ)/σ] ; Φ(−z) = 1 − Φ(z)
tables give Φ(z) for z ≥ 0- Sketch every time: shade the area you want, so you can see whether to subtract from 1.
- N(μ, σ²) has variance σ². N(50, 16) has σ = 4, not 16.
Worked example
X ~ N(50, 4²). Find (a) P(X < 56), (b) P(45 < X < 53).
- (a) z = (56 − 50)/4 = 1.5; Φ(1.5) = 0.933 M1 A1
- (b) z = −1.25 and 0.75: Φ(0.75) − [1 − Φ(1.25)] M1
- = 0.7734 − 0.1056 = 0.668 A1
2Working backwards
Inverse normal
Worked example
X ~ N(50, 4²). Find k such that P(X > k) = 0.2.
- P(X < k) = 0.8 ⇒ z = 0.842 M1
- k = 50 + 0.842 × 4 = 53.4 A1
Worked example
X ~ N(μ, σ²) with P(X < 20) = 0.1 and P(X > 32) = 0.05. Find μ and σ.
- (20 − μ)/σ = −1.282 and (32 − μ)/σ = 1.645 M1 A1 A1
- subtracting: 12/σ = 2.927 ⇒ σ = 4.10 M1
- μ = 20 + 1.282 × 4.10 = 25.3 A1
Common mistakeSigns on z. A value below the mean has a negative z. The 2024 Paper 5 report noted many candidates writing +1.282 in the first equation.
3Normal approximation to the binomial
B(n, p) ≈ N(np, npq)
- Continuity correction: a whole number r covers the interval r − 0.5 to r + 0.5. So P(X ≥ 35) becomes P(Y > 34.5), and P(X < 35) becomes P(Y < 34.5).
Worked example
X ~ B(80, 0.4). Use a suitable approximation to find P(X ≥ 35).
- np = 32, npq = 19.2; both np, nq > 5 B1
- P(Y > 34.5) with Y ~ N(32, 19.2) M1
- z = 2.5/√19.2 = 0.571 ⇒ 1 − 0.716 = 0.284 M1 A1
- (the exact binomial value is 0.283)
Common mistakeMissing or wrong-way continuity corrections, or using npq as the standard deviation. Write the inequality in X, then change it.
Linking questionThe binomial is in Discrete random variables; the normal distribution of sample means is in Sampling and estimation; and normal tests are in Hypothesis tests.
Test yourself — cover the answers first
Quick check
- Find Φ(0) for Z ~ N(0, 1).
- X ~ N(30, 9). What is σ?
- Write P(Z > −1) in terms of Φ(1).
- Continuity-correct P(X ≤ 20) for a binomial X.
- When is the normal approximation to B(n, p) suitable?
Exam-style
- Lengths are N(120, 25). Find the probability a length is within 8 of the mean.
- X ~ N(μ, 36) and P(X > 70) = 0.0228. Find μ.
- X ~ B(50, 0.5). Use the normal approximation to estimate P(X = 25).
Show answers
- 0.5
- 3
- Φ(1)
- P(Y < 20.5)
- np > 5 and n(1 − p) > 5
- z = ±1.6: 2Φ(1.6) − 1 = 0.890
- z = 2.0 ⇒ μ = 70 − 12 = 58
- N(25, 12.5): P(24.5 < Y < 25.5) = 2Φ(0.141) − 1 = 0.112
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