Edexcel A-level Further Maths revision notes

Central Limit Theorem

Section FS1-5

Year 2

1 specification point

Notes and three levels of exam-style practice for each registered specification point in this section.

Checked against Edexcel 9FM0 section FS1-5

Checked against Edexcel 9FM0 section FS1-5. Review basis: the qualification registry sourced from the Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) specification; registry verification recorded 17 July 2026.

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FS1-5.1

Applications of the Central Limit Theorem to other distributions.

Notes

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Explanation

For a large independent random sample from a population with mean $\mu$ and variance $\sigma^2$ , the Central Limit Theorem gives $\overline X\mathrel{\dot\sim}N(\mu,\sigma^2/n)$ . The standard error is $\sigma/\sqrt n$ , so standardise with $Z=(\overline X-\mu)/(\sigma/\sqrt n)$ .
Applications may use any distribution studied in A-level Mathematics or Further Statistics 1; first obtain that population's mean and variance.
No proof is required.
The approximation improves with sample size and generally needs more caution for strongly skewed populations.
Examiners expect the variance of the sample mean to be divided by $n$ , approximation notation, and no continuity correction when the sample mean is treated directly.

The approximate normal distribution of a large-sample mean, centred at the population mean.

Worked example

A population has mean $30$ and variance $64$ . For a random sample of size $100$ , estimate $P(28.8<\overline X<31.6)$ .

1. $\overline X\mathrel{\dot\sim}N(30,64/100)$ , so the standard error is $0.8$ .
2.The lower standardised value is $(28.8-30)/0.8=-1.5$ .
3.The upper standardised value is $(31.6-30)/0.8=2$ .
4. $P(-1.5<Z<2)=0.9104$ to $4$ significant figures.

Answer: The estimated probability is $0.9104$ .

Common mistakes

Don't use variance $64$ for $\overline X$ instead of $64/100$ .
Don't divide the standard deviation by $n$ instead of by $\sqrt n$ .
Don't add a continuity correction to bounds already stated for the sample mean.

Exam tip

Write the approximate distribution of $\overline X$ before standardising either boundary.

Tier 1 · Easy

ORIGINAL

Independent lifetimes have population mean

50

hours and variance

100

hours

^2

. For a sample of

36

lifetimes, use the Central Limit Theorem to estimate

P(\overline X<47.5)

(4)

(Total for Question 1 is 4 marks)

Tier 2 · Standard

ORIGINAL

A geometric population has parameter

p=0.32

. A random sample of

45

observations is taken. Find the value of

c

for which

P(\overline X>c)\approx0.05

, giving

c

3

decimal places.

(6)

(Total for Question 1 is 6 marks)

Tier 3 · Hard

ORIGINAL

A population has mean

\mu

and variance

36

. Using the Central Limit Theorem normal approximation, estimate the least sample size

n

for which

P(|\overline X-\mu|<1.2)\geq0.95

(6)

(Total for Question 1 is 6 marks)

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