Contents · 0% complete

Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9FM0Option · Further Statistics 2Estimation, confidence intervals and testsSampling and unbiased estimates

Estimation, confidence intervals and tests · Subtopic 1 of 3

Sampling and Unbiased Estimates

We usually can't measure a whole population, so we measure a random sample and estimate. The sample mean estimates $\mu$ fairly, but for the variance we divide by $n-1$ , not $n$ , to stop the estimate coming out too small on average.

Cambridge 9709 Paper 6Edexcel YMA01 WST03Edexcel 9FM0 Option≈ 35 minRevision 5 · updated 2026-10-27

Practise this topicTimed test Ask Salam TutorSwitch board

Section 1 of 14

Learning objectives

Distinguish a population from a sample, and a parameter from a statistic.
Explain why samples should be random, and criticise a biased method.
Calculate unbiased estimates $\bar x$ and $s^2=\dfrac{1}{n-1}\left(\Sigma x^2-\dfrac{(\Sigma x)^2}{n}\right)$ .
Use coded totals to find unbiased estimates.

Pause and think

To estimate how many hours a week students exercise, you ask the first $20$ people you meet at the gym. What is wrong with this sample?

Section 2 of 14

Core idea

Start hereTasting the soup

A chef tastes one spoonful to judge the whole pot. That works only if the soup is stirred: the spoonful must be representative. Statistics does the same with populations: study a well-chosen sample, then estimate.

The big idea

A random sample avoids bias. Unbiased estimates: $\bar x$ for $\mu$ ; $s^2=\frac1{n-1}\left(\sum x^2-\frac{(\sum x)^2}n\right)$ for $\sigma^2$ (divide by $n-1$ , not $n$ ).

1. Sampling

A random sample of 6 from a population

Random sampling gives every member an equal chance of being chosen, so the sample is not biased towards one type.

Population: everyone of interest. Census: measuring them all.
Random sample: every member equally likely; random numbers.
Bias: a method that systematically favours some members.

2. Why n − 1?

The sample variance underestimates

$\sum(x-\bar x)^2\le\sum(x-\mu)^2$
$\bar x$ is the value that minimises the sum of squared deviations from the sample.
$\text{so }\tfrac1n\sum(x-\bar x)^2\text{ tends to be too small}$
It uses the sample's own centre.
$E\left[\sum(x-\bar x)^2\right]=(n-1)\sigma^2$
(A standard result.)
$s^2=\frac{1}{n-1}\sum(x-\bar x)^2\ \Rightarrow\ E(s^2)=\sigma^2$
Divide by $n-1$ for an unbiased estimate.

Worked exampleSample $4,7,8,9,12$

Unbiased estimates of the mean and variance.

Goal: $\bar x$ , $s^2$ .
Strategy: Sums.

$\sum x=40,\ \sum x^2=354$
Totals.
$\bar x=8$
Mean.
$s^2=\tfrac14\left(354-\tfrac{1600}5\right)=\tfrac{34}4=8.5$
Formula.

Answer

8

;

8.5

✓ CheckDeviations

-4,-1,0,1,4

: squares sum

34

✓.

Pause and think

Why is surveying gym-goers poor for "average exercise of adults"?

Summary

Random samples.
$\bar x$ ; $s^2$ with $n-1$ .

Unbiased estimate: Correct on average.

HistoryThe poll that picked the wrong president. In 1936 the Literary Digest magazine polled over two million people and predicted Alf Landon would beat Franklin Roosevelt easily. Roosevelt won in a landslide. The sample came largely from telephone directories and car registrations, which in the Depression meant wealthier voters: a huge but biased sample. George Gallup's much smaller, better-chosen sample got it right.

Fun factBessel's correction. Dividing by

n-1

is called Bessel's correction, after the astronomer Friedrich Bessel. One way to see it: once you know

\bar x

, only

n-1

of the deviations

x-\bar x

are free to vary (they must add to zero), so there are only

n-1

"degrees of freedom".

Section 3 of 14

Key formulae

Unbiased estimate of μ

\bar x=\frac{\Sigma x}{n}

Unbiased estimate of σ²

s^2=\frac{1}{n-1}\left(\Sigma x^2-\frac{(\Sigma x)^2}{n}\right)=\frac{n}{n-1}\left(\frac{\Sigma x^2}{n}-\bar x^2\right)

With coded totals

s^2=\frac{1}{n-1}\left(\Sigma(x-a)^2-\frac{(\Sigma(x-a))^2}{n}\right)

TrickTwo-step safety net. Work out the ordinary (divide by

n

) variance first, as in Paper 5, then multiply by

\dfrac{n}{n-1}

. It uses the formula you already know and makes the "

n-1

" step impossible to forget.

Section 4 of 14

Worked examples

Example 1 · From totals0 / 2 steps

A sample of

8

values has

\Sigma x=96

and

\Sigma x^2=1200

. Find unbiased estimates of the population mean and variance.

Example 2 · Coded totals0 / 2 steps

For

20

values,

\Sigma(x-10)=30

and

\Sigma(x-10)^2=225

. Find unbiased estimates of the mean and variance.

Example 3 · Critiquing a sample0 / 1 steps

A council surveys residents' views on a new car park by asking people in the town centre on a weekday morning. Why may this be unsatisfactory?

Section 5 of 14

Explore the graph

How much does n − 1 change the estimate?

Loading the interactive graph…

Try this

The curve is the factor $\dfrac{n}{n-1}$ that turns the sample variance into the unbiased estimate.
For a sample of $2$ , the correction doubles the variance. For $50$ , it adds only $2\%$ .
Why does the correction matter most for small samples?

For large samples, dividing by $n$ or $n-1$ makes little difference; for small ones it matters a lot.

In the real worldTesting to destruction. Car makers crash-test a small random sample of cars, and food companies test a random sample of tins for contamination. Testing everything would destroy the whole product, so estimation from samples is not just convenient but essential.

Section 6 of 14

Common mistakes

IncorrectDividing by

n

for an unbiased variance

CorrectDivide by

n-1

Dividing by

n

underestimates

\sigma^2

on average.

Incorrect

s^2=\dfrac{\Sigma x^2}{n-1}-\bar x^2

Correct

\dfrac1{n-1}\left(\Sigma x^2-n\bar x^2\right)

The whole bracket is divided by

n-1

Incorrect"A large sample is always unbiased"

CorrectRandomness matters, not size

The Literary Digest sample was huge and biased.

IncorrectForgetting to add

a

back with coded data

Correct

\bar x=a+\dfrac{\Sigma(x-a)}n

Only the variance ignores

a

the sample variance is $\dfrac{\sum(x-\bar x)^2}{n}$

Unbiased estimate

divide by $n-1$

why? Sample: because I'm always a little too close to my own mean

Section 7 of 14

Exam tip

Exam tipCriticise in context. "Explain why the sample may be unsatisfactory" needs a reason linked to the situation: who is left out, and why they might be different. "It is not random" alone usually scores nothing.

Section 8 of 14

Guided practice

SA-9709-P6-SE-G013 marksL1 FoundationCalculatorUnbiased estimates

A sample of

10

values gives

\Sigma x=52

and

\Sigma x^2=290

. Find unbiased estimates of the population mean and variance (variance to 3 s.f.).

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-9709-P6-SE1-0013 marksL1 FoundationCalculatorUnbiased estimates

A sample of

8

values has

\Sigma x=96

and

\Sigma x^2=1200

. Find unbiased estimates of the population mean and variance (variance to 3 s.f.).

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-SE1-0053 marksL2 StandardCalculatorUnbiased estimates

A random sample of

5

values is

3,\ 5,\ 6,\ 8,\ 8

. Find unbiased estimates of the population mean and variance.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-SE1-0021 markL1 FoundationCalculatorUnbiased estimates

A sample of

5

values has variance

4

(calculated with divisor

n

). Find an unbiased estimate of the population variance.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-SE-0013 marksL2 StandardCalculatorUnbiased estimates

A sample of

50

values gives

\Sigma(x-20)=35

and

\Sigma(x-20)^2=1250

. Find unbiased estimates of the population mean and variance (variance to 3 s.f.).

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-SE1-0041 markL1 FoundationCalculatorSampling

To estimate how often students exercise, a researcher questions the first

20

students at the gym. The main problem is that

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-SE-0071 markL1 FoundationCalculatorSampling

Why is a random sample used?

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-9709-P6-SE1-0074 marksL3 ExamCalculatorUnbiased estimates

For a sample of

20

values,

\Sigma(x-10)=30

and

\Sigma(x-10)^2=225

. Find unbiased estimates of the population mean and variance (variance to 3 s.f.).

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Random samples avoid bias; size alone does not.
Unbiased estimate of $\mu$ : $\bar x$ .
Unbiased estimate of $\sigma^2$ : divide by $n-1$ .
Coded totals work the same way.

Section 12 of 14

Quick quiz

Q1 · SA-9709-P6-SE-Q031 markL1 FoundationCalculatorSampling

A sample is used rather than a census because

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-9709-P6-SE-0101 markL1 FoundationCalculatorUnbiased estimates

An unbiased estimate of population variance from a sample of size

n

divides by

Salam Academy Original Practice · Proposed mark scheme

Q3 · SA-9709-P6-SE1-0031 markL1 FoundationCalculatorSampling

Which is a way of taking a random sample of

20

students from a school of

800

Salam Academy Original Practice · Proposed mark scheme

Q4 · SA-9709-P6-SE1-0061 markL1 FoundationCalculatorSampling

In statistics, the "population" is

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Mathematics
Where: WST03 (Statistics 3) · Sampling, estimation and confidence intervals
Expected: Populations, samples, sampling frames and sampling methods (WST03), with unbiased estimators of μ and σ².
Calculator: Permitted. Statistical tables are provided in the formula booklet.
Typical style: Includes stratified and quota sampling, and confidence intervals for the difference of two means.
Open in: YMA01 · WST03 9FM0 · Option

Section 14 of 14

Past paper questions

4 questions (23 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

9709/62 · May/June 2025Question 16 marks
Use a dice to choose at random, then the probability that a difference of two normal times exceeds a value.
Question paper, page 3 Mark scheme
9709/62 · May/June 2025Question 23 marks
Find a sample mean from data using an unbiased estimate of variance.
Question paper, page 4 Mark scheme
9709/62 · May/June 2024Question 26 marks
Random sampling, unbiased estimates of mean and variance.
Question paper, page 3 Mark scheme
9709/62 · May/June 2023Question 38 marks
Unbiased estimates from summary data, then a test of a mean at 1%.
Question paper, page 4 Mark scheme

All 9709 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel 9FM0, First teaching 2017 (official specification).