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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
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Summary
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LearnYMA01WST02 · Statistics 2Hypothesis testsPopulations and sampling methods

Hypothesis tests · Subtopic 2 of 3

Statistical Sampling

A census measures every member of a population; a sample measures some, so it is cheaper and quicker but may be biased. Random methods (simple random, systematic, stratified) need a sampling frame; non-random methods (quota, opportunity) don't, but are more prone to bias.

Edexcel YMA01 WST02Edexcel YMA01 WST03Edexcel 9MA0 Paper 3AQA 7357 Paper 3≈ 30 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Define population, census, sample, sampling unit and sampling frame.
Describe simple random, systematic, stratified, cluster, quota and opportunity sampling.
Choose and criticise a sampling method for a context.

Pause and think

A school wants to know how students travel in. Why not just ask the first 30 students through the gate?

Section 2 of 14

Core idea

Start hereThe poll that got it wrong

In 1936 a magazine asked 2.4 million Americans how they'd vote and confidently predicted the wrong winner. A modern poll of about 1000 people would have got it right. How you choose a sample matters far more than how big it is.

The big idea

A sample should represent its population. Random methods give every member a known chance of selection and need a sampling frame; non-random methods are quicker but risk bias. Bigger samples reduce random error, not bias.

1. Key terms

Population: everything of interest. Census: data from every member.
Sample: a subset. Sampling unit: one member. Sampling frame: a list of the units.

\text{stratum sample}=\frac{\text{stratum size}}{\text{population size}}\times n

$n$: total sample size.

2. Methods

Worked exampleStratified sample

A company has 120 managers, 300 office staff and 780 factory workers. Take a stratified sample of 80.

Goal: Numbers from each group.
Strategy: Proportional allocation.

$N=1200$
Population size.
$\tfrac{120}{1200}\times80=8,\ \tfrac{300}{1200}\times80=20$
Managers and office staff.
$\tfrac{780}{1200}\times80=52$
Factory workers.

Answer

8

20

52

; then choose randomly within each group.

✓ Check

8+20+52=80

✓.

Worked exampleSystematic sample

Describe a systematic sample of 20 from a numbered list of 400.

Goal: A method.
Strategy: Interval $k=\frac Nn$ .

$k=\frac{400}{20}=20$
Interval.
$\text{random start }r\in\{1,\dots,20\}$
Randomise the start, e.g. with a calculator.
$r,\ r+20,\ r+40,\dots$
Every 20th item.

Answer20 items, evenly spaced from a random start.

✓ CheckLargest index

r+380\le400

✓.

Choosing a method

Drawing diagram…

Start by asking whether you have a list of the population.

Common misconceptionSize isn’t everything A huge biased sample is still biased. Randomisation, not size, removes bias.

Key takeawayWhen a census is impossible Destructive testing (e.g. light-bulb lifetimes) or huge populations make a census impractical.

Pause and think

Why might systematic sampling go wrong on a list of houses where every 10th is a corner house?

Summary

Census vs sample.
Random: simple, systematic, stratified (need a frame).
Non-random: quota, opportunity.
Stratified sizes are proportional.

Sampling frame: List of the population used to select a sample.
Bias: Systematic distortion of results.
Stratum: A distinct subgroup of the population.

HistoryThe 1936 poll. The Literary Digest polled 2.4 million people and predicted Landon would beat Roosevelt. Their lists came from car and phone owners, a wealthy group: Roosevelt won by a landslide. Size can't fix bias.

Fun factTasting soup. You can judge a whole pan of soup from one spoonful, as long as you stir first. Stirring is randomisation.

Section 3 of 14

Key formulae

Systematic

k=\frac{N}{n};\ \text{random start in }1..k,\ \text{then every }k\text{th}

Stratified

\text{sample from stratum}=\frac{\text{stratum size}}{N}\times n

TrickRandom needs a list. If there's no sampling frame (a list of the population), you can't do simple random, systematic or stratified sampling: use quota or opportunity.

Section 4 of 14

Worked examples

Example 1 · Stratified0 / 2 steps

A school has 360 boys and 240 girls. Find the numbers for a stratified sample of 50.

Example 2 · Systematic0 / 2 steps

Describe a systematic sample of 20 from a list of 400.

Section 5 of 14

Explore the graph

Bigger samples, smaller spread

Loading the interactive graph…

Try this

$x$ is sample size; $y$ is the typical error of a sample mean, $\frac{\sigma}{\sqrt n}$ .
Quadrupling the sample only halves the error, and does nothing to remove bias.

Size reduces random error, not bias.

In the real worldOpinion polls. Modern polls use quota sampling by age, sex and region, then weight results: about 1000 people can estimate national opinion to within about 3 percentage points.

Section 6 of 14

Common mistakes

Incorrect"Random" meaning haphazard

CorrectEvery member equally likely

Picking people you happen to meet is opportunity sampling.

IncorrectStratified needs equal numbers per group

CorrectProportional to group size

That's what keeps it representative.

IncorrectLarger sample removes bias

CorrectOnly random error shrinks

A biased method stays biased.

Survey of my friends

100% agree with me. Opportunity sampling strikes again.

Section 7 of 14

Exam tip

Exam tipName the frame. When describing a method, say what the sampling frame is (e.g. the school register) and how you'd randomise (random number generator).

Section 8 of 14

Guided practice

SA-ST-SA-G012 marksL1 FoundationNon-calculatorMethods

A school has 360 boys and 240 girls. How many boys should be in a stratified sample of 50?

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-ST-SA-0011 markL1 FoundationNon-calculatorMethods

For a systematic sample of 20 from a list of 400, what is the interval

k

Salam Academy Original Practice · Proposed mark scheme

SA-ST-SA-0022 marksL2 StandardNon-calculatorMethods

A company has 120 managers, 300 office staff and 780 factory workers. How many factory workers belong in a stratified sample of 80?

Salam Academy Original Practice · Proposed mark scheme

SA-ST-SA-0031 markL1 FoundationNon-calculatorKey terms

A list of every member of the population from which a sample is drawn is called

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

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Section 11 of 14

Summary

Census vs sample.
Random methods need a frame.
Stratified: proportional.
Size doesn't fix bias.

Section 12 of 14

Quick quiz

1 more quiz question with hints, worked solutions and instant marking
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Section 13 of 14

Same topic, different board

Specification: A level Mathematics 9MA0
Where: Paper 3 · Statistical sampling
Expected: Population, sample, census, sampling unit and frame; simple random, systematic, stratified, quota and opportunity sampling; the large data set.
Calculator: Permitted.
Typical style: Often tied to the large data set: explain why a method suits a given context.
Open in: YMA01 · WST02 YMA01 · WST03 9MA0 · Paper 3

Section 14 of 14

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel YMA01, 2018 specification (official specification).