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Contents · 0% complete
  1. Learning objectives
  2. Core idea
  3. Key formulae
  4. Worked examples
  5. Explore the graph
  6. Common mistakes
  7. Exam tip
  8. Guided practice
  9. Practice questions
  10. Challenge
  11. Summary
  12. Quick quiz
  13. Same topic, different board
  14. Past paper questions
Learn7357Paper 3 · Pure and StatisticsStatistical samplingPopulations and sampling methods
Statistical sampling · Subtopic 1 of 1

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 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.
stratum sample=stratum sizepopulation size×n\text{stratum sample}=\frac{\text{stratum size}}{\text{population size}}\times n
where
nn
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.
  1. N=1200N=1200
    Population size.
  2. 1201200×80=8, 3001200×80=20\tfrac{120}{1200}\times80=8,\ \tfrac{300}{1200}\times80=20
    Managers and office staff.
  3. 7801200×80=52\tfrac{780}{1200}\times80=52
    Factory workers.
Answer88, 2020, 5252; then choose randomly within each group.
✓ Check8+20+52=808+20+52=80 ✓.
Worked exampleSystematic sample

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

Goal
A method.
Strategy
Interval k=Nnk=\frac Nn.
  1. k=40020=20k=\frac{400}{20}=20
    Interval.
  2. random start r∈{1,…,20}\text{random start }r\in\{1,\dots,20\}
    Randomise the start, e.g. with a calculator.
  3. r, r+20, r+40,…r,\ r+20,\ r+40,\dots
    Every 20th item.
Answer20 items, evenly spaced from a random start.
✓ CheckLargest index r+380≤400r+380\le400 ✓.
Choosing a method

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.
Key words
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

Systematick=Nn; random start in 1..k, then every kthk=\frac{N}{n};\ \text{random start in }1..k,\ \text{then every }k\text{th}
Stratifiedsample from stratum=stratum sizeN×n\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
      1. xx is sample size; yy is the typical error of a sample mean, σn\frac{\sigma}{\sqrt n}.
      2. 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?

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      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 kk?

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      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?

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      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
      Choose one option

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      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-ST-SA-0041 markL2 StandardNon-calculatorEvaluating
      An interviewer must question 20 men and 20 women in a shopping centre, choosing whoever is willing. This is
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      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

      Q1 · SA-ST-SA-Q011 markL1 FoundationNon-calculatorEvaluating
      Which is an advantage of a census over a sample?
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      Section 13 of 14

      Same topic, different board

      Specification
      7357 A-level Mathematics
      Where
      Paper 3 · Statistical sampling
      Expected
      Population and sample, census, sampling frame; simple random, systematic, stratified, cluster, quota and opportunity sampling; advantages and disadvantages; selecting and critiquing methods.
      Calculator
      Permitted.
      Typical style
      Short-answer questions: "describe how to take a stratified sample" or "give one disadvantage of …".
      Open in
      7357 · Paper 3
      Section 14 of 14

      Past paper questions

      No past paper questions are tagged to this subtopic yet. We add papers every week.

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      Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 7357, First teaching 2017 (official specification).