S1StatisticsFoundation & HigherGrades 3–6

Sampling

A school wants to know how its 1200 students travel in. Asking everyone takes too long, so it asks 60. Whether those 60 answers say anything useful about the whole school depends entirely on how the 60 were chosen. That is the study of sampling.

35 min AQA GCSE Maths

The big picture

The $\textbf{population}$ is everyone or everything you want to know about; a $\textbf{sample}$ is the part you actually collect data from. A good sample represents the population fairly, so its results can be used to make inferences.

A sample is $\textbf{biased}$ when some members of the population are more likely to be chosen than others. Random sampling removes that, and larger samples make estimates more reliable — although no sample is ever guaranteed to match the population exactly.

What you'll learn

Define population and sample
Identify sources of bias in a sampling method
Describe how to take a simple random sample
Use a sample to make inferences about a population, and explain limitations

Populations and samples

A $\textbf{census}$ collects data from the whole population. It is accurate but can be slow, expensive, or impossible — testing every light bulb until it fails would destroy them all.

A sample is quicker and cheaper, but only gives an estimate.

Tip — When asked for an advantage of a sample, say "quicker" or "cheaper". For a disadvantage, say "may not represent the whole population".

Bias

Asking the first 60 students through the gate: late-comers and bus users arriving together may be missed or over-represented.

Asking only Year 7: the sample does not include other year groups.

Surveying people at a gym about exercise habits: they are more active than the general population.

Small samples can also be unrepresentative purely by chance.

Example 1A council asks people in a town-centre car park whether a new car park is needed. Explain why the sample may be biased.

1Everyone asked has driven into town.

2People who walk, cycle or use buses are left out.

AnswerDrivers are more likely to want more parking, so the results will over-represent support for a new car park.

Random sampling

In a $\textbf{simple random sample}$ , every member of the population has an equal chance of being chosen.

Method: number every member of the population, then use a random number generator to pick numbers, ignoring repeats, until the sample is the right size.

Your turnSame method, new numbers — try it before moving on.

A school has 1,200 students, of whom 240 are in Year 7. A stratified sample of 60 students is chosen. How many should be from Year 7?

Your answer

Show worked solution

Solution

1 $\frac{240}{1200} = \frac{1}{5}$ of the school is in Year 7.
2 $\frac{1}{5} \times 60 = 12$ .

Answer: $12$

Why this works

Random does not mean "whoever is nearby". Asking people you happen to meet is convenient, not random, and it is one of the most common sources of bias.

Making inferences

Scale up sample proportions to estimate population values.

A different sample would give slightly different results, so any estimate carries uncertainty. A larger random sample reduces it.

Example 1In a random sample of 60 of the 1200 students, 21 walk to school. Estimate how many students in the school walk.

1Proportion:

\dfrac{21}{60} = 0.35

0.35 \times 1200

AnswerAbout 420 students

Your turnSame method, new numbers — try it before moving on.

In a random sample of 50 of the 900 workers at a company, 18 are vegetarian. Estimate how many workers at the company are vegetarian.

Your answer

Show worked solution

Solution

1Proportion in the sample: $\frac{18}{50} = 0.36$ .
2 $0.36 \times 900 = 324$ .

Answer: $324$

Tip — Say "about" or "estimate" — the sample cannot give the exact figure.

Think like an examiner

Explain bias in context — say who is left out or over-represented.
Describe random sampling in steps: number, generate, select.
Comment on sample size when judging reliability.

Remember these

\text{estimate} = \dfrac{\text{sample count}}{\text{sample size}} \times \text{population}

Scaling up from a sample.

\text{group sample} = \dfrac{\text{group size}}{\text{population}} \times \text{sample size}

Sampling groups in proportion.

n \uparrow \Rightarrow \text{reliability} \uparrow

Larger samples are more reliable.

Watch out for these

Saying a sample is biased "because it is small".

Small samples are less reliable; bias comes from the selection method.

Describing random sampling as "picking people at random in the street".

Number the population and use random numbers.

Stating an inference as exact.

It is an estimate.

Defining the population as the sample.

The population is the whole group being studied.

Stretch yourself

A gym has 450 members: 180 are under 30, 200 are aged 30–59 and 70 are 60 or over. The manager wants a sample of 50 that reflects the age groups. (a) How many from each group should be chosen? (b) In the sample, 12 of the under-30s say they would use a new spin studio. Estimate how many under-30 members in the gym would use it.

Hint — Each group’s share of the sample should match its share of the membership.

Reveal solution

Solution(a) Under 30:

\dfrac{180}{450} \times 50 = 20

. Aged 30–59:

\dfrac{200}{450} \times 50 = 22.2 \approx 22

. 60 or over:

\dfrac{70}{450} \times 50 = 7.8 \approx 8

. Check:

20 + 22 + 8 = 50

. (b)

\dfrac{12}{20} = 0.6

of the sampled under-30s, so about

0.6 \times 180 = 108

under-30 members.

\blacksquare

Questions students ask

Is a bigger sample always better?

It is more reliable, but a large biased sample is still biased.

What is a sampling frame?

The list of population members you choose from, such as a school register.

Can a random sample be unrepresentative?

Yes, by chance — but it is not biased, and larger random samples reduce the risk.

Key takeaways

The population is the whole group; a sample is part of it.
Bias happens when some members are more likely to be chosen.
A simple random sample gives everyone an equal chance.
Scale up sample proportions to estimate population values.
Larger random samples give more reliable estimates.

Exam-style questions

Practise it the way it is examined

3 original questions · 6 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 31 mark

Jo wants to find out how much exercise people in her town do. She asks people at the local gym. Give one reason why her sample may be biased.[1]
Show mark scheme
- B1People at a gym are likely to exercise more than the population as a whole

Question 2Grade 52 marks

A company has 400 workers: 120 office staff, 200 factory staff and 80 drivers. A stratified sample of 50 workers is taken.

How many of each type of worker should be in the sample?[2]
Show mark scheme
- M1Uses $\frac{50}{400}$ or equivalent for at least one group
- A1Office 15, factory 25, drivers 10

Question 3Grade 63 marks

60 fish in a lake are caught, tagged and released. A week later, 50 fish are caught and 8 of them are tagged.

Estimate the number of fish in the lake, and state one assumption you have made.[3]
Show mark scheme
- M1 $\frac{8}{50} = \frac{60}{N}$
- A1 $375$
- B1e.g. the population does not change, or tagged fish mix evenly
Answer: $375$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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