1MA1 · S1StatisticsFoundation & HigherGrades 3–6

Sampling

To find out what the whole school thinks about the canteen, you could ask every student — or ask a well-chosen $\textbf{sample}$ and draw conclusions about the $\textbf{population}$ . Sampling saves time and money, but only if the sample is representative. Choosing badly produces $\textbf{bias}$ , and no amount of careful calculation afterwards can fix it.

35 min Statistics

The big picture

Every statistical conclusion depends on how the data were collected. A survey of people leaving a gym will overestimate how much the town exercises; a phone poll during working hours misses people at work. Spotting and explaining bias like this is a regular Edexcel question.

At Higher tier, capture–recapture uses a sample in a clever way: mark some animals, release them, and use the proportion marked in a later sample to estimate the total population.

What you'll learn

Distinguish between a population and a sample
Identify sources of bias and suggest improvements
Describe random and stratified sampling
Estimate population size using capture–recapture (Higher)

Populations and samples

The $\textbf{population}$ is the whole group being studied. A $\textbf{sample}$ is part of it. A census surveys everyone.

Samples are cheaper and quicker, but give less certain conclusions. Larger samples are generally more reliable.

A sample is $\textbf{representative}$ if it reflects the population’s make-up.

Bias

A sample is $\textbf{biased}$ if some members of the population are more likely to be included than others, in a way that affects the results.

Common causes: sampling at a particular time or place, using volunteers, too small a sample, or leading questions.

To reduce bias, use a random method, a larger sample, and ask neutral questions.

Example 1To find the favourite sport of students in a school, Jo asks the first 20 students arriving at football practice. Explain why the sample is biased.

1Everyone surveyed plays football, so football is over-represented.

2Only 20 students are asked, which is a small sample.

AnswerThe sample over-represents football players and is too small; a random sample across the whole school would be better.

Tip — When explaining bias, refer to the specific context: say who is over- or under-represented and why.

Random and stratified sampling

In a $\textbf{simple random sample}$ , every member of the population has an equal chance of selection — for example, numbering everyone and using a random number generator.

In a $\textbf{stratified sample}$ , the population is split into groups (strata), and each group is sampled in proportion to its size.

Number from each group $= \dfrac{\text{group size}}{\text{population size}} \times \text{sample size}$ .

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

Stratified sampling.

Example 1A school of 1200 students has 300 in Year 7. A stratified sample of 60 is taken. How many Year 7 students should be included?

\dfrac{300}{1200} \times 60

Answer15 students

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

A club has 900 members, of whom 180 are juniors. A stratified sample of 50 is taken. How many juniors should be in it?

Your answer

Show worked solution

Solution

1 $\frac{180}{900} = \frac{1}{5}$ .
2 $\frac{1}{5} \times 50 = 10$ .

Answer: $10$

Capture–recapture

Capture a sample, mark them and release them. Later, capture a second sample and count how many are marked.

Assume the proportion marked in the second sample equals the proportion marked in the whole population: $\dfrac{M}{N} = \dfrac{m}{n}$ .

So the population estimate is $N = \dfrac{M \times n}{m}$ , where $M$ is the number first marked, $n$ the size of the second sample and $m$ the number marked in it.

N \approx \dfrac{Mn}{m}

Capture–recapture estimate.

Example 140 fish are caught, marked and released. Later 50 fish are caught, of which 8 are marked. Estimate the number of fish in the lake.

\dfrac{40}{N} = \dfrac{8}{50}

N = \dfrac{40 \times 50}{8}

AnswerAbout 250 fish

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

30 fish are caught, marked and released. Later, 40 fish are caught and 6 are marked. Estimate the population.

Your answer

Show worked solution

Solution

1 $\frac{6}{40} = \frac{30}{N}$ .
2 $N = \frac{30 \times 40}{6} = 200$ .

Answer: $200$

Why this works

The method assumes the population does not change between samples, that marks are not lost, and that marked animals mix back in randomly. If any assumption fails, the estimate is unreliable — and Edexcel often asks you to name one.

Think like an examiner

Explain bias in context, naming who is over- or under-represented.
Round stratified sample numbers sensibly to whole people.
State an assumption when using capture–recapture.

Sampling

\tfrac{\text{group}}{\text{population}} \times \text{sample}

Stratified sample size.

N = \tfrac{Mn}{m}

Capture–recapture.

Watch out for these

Saying a sample is biased "because it is not random" with no context.

Explain how the method favours particular groups.

Dividing the population by the sample for stratified numbers.

Multiply the group’s fraction of the population by the sample size.

Using

N = \tfrac{Mm}{n}

N = \tfrac{Mn}{m}

: marked first × second sample ÷ marked in second.

Assuming a bigger sample removes all bias.

A large biased sample is still biased.

Stretch yourself

A scientist marks 60 birds and releases them. A week later she catches 75 birds, of which 9 are marked. Estimate the population, and explain one reason why the actual population might be larger than the estimate.

Hint — Use $N = \dfrac{Mn}{m}$ . Then think about what happens if marked birds leave the area.

Reveal solution

Solution

N = \dfrac{60 \times 75}{9} = 500

birds. The real population could be larger if marked birds are easier to catch again — for example, if they learned where the traps are, or had not yet mixed back into the whole population. Then more marked birds appear in the second sample than they should, so

m

is too large and

N = \tfrac{Mn}{m}

comes out too small.

\blacksquare

Questions students ask

Why not always use a census?

A census is often too expensive or slow, and sometimes impossible — testing every light bulb to destruction would leave none to sell.

What makes a sample random?

Every member of the population has an equal chance of being chosen, usually using a random number generator.

When is stratified sampling better?

When the population has distinct groups that might answer differently, such as year groups or age bands.

Key takeaways

A sample is part of a population; a census is all of it.
Bias occurs when the sampling method favours some members.
Random samples give every member an equal chance.
Stratified samples take each group in proportion to its size.
Capture–recapture: $N = \tfrac{Mn}{m}$ , with assumptions about the population.

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

Sam surveys people outside a gym about how often they exercise. Explain why the sample may be biased.[1]
Show mark scheme
- B1People at a gym probably exercise more than average

Question 2Grade 52 marks

A school has 280 Year 9, 300 Year 10 and 220 Year 11 students. A stratified sample of 40 is taken.

How many students from each year should be chosen?[2]
Show mark scheme
- M1Uses $\frac{40}{800}$
- A1Year 9: 14, Year 10: 15, Year 11: 11

Question 3Grade 63 marks

50 animals are tagged. Later 60 are caught, of which 12 are tagged. Estimate the population and state one assumption.[3]
Show mark scheme
- M1 $\frac{12}{60} = \frac{50}{N}$
- A1 $250$
- B1e.g. the population is unchanged between samples
Answer: $250$

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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