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GCSE Maths · Statistics
Sampling GCSE Questions and Worked Answers
A sample is a smaller group studied to learn about a population. A useful sample should represent the population, not just be large. Random selection reduces selection bias; proportional allocation can preserve known subgroup shares. Conclusions remain estimates.
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Start with the meaning
What you need to know about sampling
Suppose you want to know how all 600 pupils at a school travel there. Asking everyone would be a census. Asking a smaller group, perhaps 60 pupils, gives a sample. The population is the whole group you want to learn about; the sample is the group you actually collect data from.
See the idea first
A smaller group should still reflect the larger one
A sample is biased when its selection systematically favours some kinds of people or results. Asking only pupils at the bike racks would over-represent cyclists. A simple random sample uses a complete list and gives each pupil an equal chance of selection. A proportional sample first divides the population into groups, called strata, and allocates sample places in their population proportions.
| Group | Population count | Share of 200 | Sample allocation |
|---|---|---|---|
| Junior | 80 | 80/200 = 40% | 16 |
| Senior | 120 | 120/200 = 60% | 24 |
| Total | 200 | 100% | 40 |
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Describe random selection
What the problem asks: Choose 30 pupils from a school list of 600.
How to solve it: Number everyone 1–600, use a random-number generator to choose 30 distinct numbers, and survey those pupils. Ignore duplicate selections; do not replace nonresponders with convenient friends.
Allocate a proportional sample
What the problem asks: A club has 80 juniors and 120 seniors. Allocate a sample of 40.
How to solve it: Juniors receive (80/200) × 40 = 16 places and seniors 24. Randomly choose distinct members within each group.
Evaluate a survey
What the problem asks: A transport survey is completed only by volunteers on a cycling website. Can it estimate the national proportion who cycle?
How to solve it: Not reliably. The sample favours people interested in cycling and willing to respond; increasing that same sample's size does not fix the selection bias.
Scale a sample result to the population
What the problem asks: In a representative sample of 50 from 600 pupils, 15 cycle. Estimate how many pupils in the school cycle.
How to solve it: The observed share is 15/50 = 0.3. If the population has approximately the same share, 0.3 × 600 = 180 pupils cycle. This is an estimate: the sample must represent the population, and random variation remains.
Estimate a population by capture–recapture (Edexcel Higher)
What the problem asks: Mark and release 40 fish. Later catch 50 fish, of which 10 are marked. Estimate the population.
How to solve it: Let N be the total number of fish. If the marked fraction in the second catch represents the whole population, 10/50 is approximately 40/N. Thus N ≈ 40 × 50 ÷ 10 = 200 fish. This needs a closed population, retained marks, thorough mixing and similar capture chances; it is an estimate, not an exact count.
A reliable routine
Allocate and select a proportional sample
Use this specific method when subgroup sizes are known and the question asks for a sample preserving those proportions. Each subgroup should contribute the same fraction of the sample as of the population. Other tasks may instead call for a simple random sample or an evaluation, not this calculation.
- Add subgroup counts to find the population total.
- For each subgroup, multiply its population fraction by the required sample size.
- Check integer allocations add to the required total; resolve rounding as the task instructs.
- Select randomly within each subgroup and report non-response or other limitations.
Check: Random does not guarantee a perfectly representative result in one sample. Larger well-selected samples usually reduce chance variation, but size alone cannot correct a biased selection method.
Fully worked
Sampling GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
Identify the groups
Question
A researcher asks 80 of a town's 4000 residents about bus use, aiming to describe all town residents. Identify the population and sample.
Population: all 4000 town residents. Sample: the 80 residents actually asked. The intended conclusion concerns the population, not only the respondents.
Example 2
Simple random method
Question
Describe how to select 25 different students at random from a complete list of 500.
Assign each student a unique number 1–500. Use a random-number generator to select 25 distinct numbers in that range, ignoring duplicates. Survey the corresponding students. Using the complete list gives every student a selection chance; choosing the first 25 names is not random.
Example 3
Proportional allocation
Question
A club has 90 juniors and 60 seniors. Allocate a proportional sample of 30.
Total population = 150.
Check 18 + 12 = 30. Then choose randomly within each group.
Example 4
Three groups
Question
A company has 120 office, 80 warehouse and 40 delivery staff. Allocate a proportional sample of 60.
Total = 240, so sample fraction = 60/240 = 1/4.
Select 30 office, 20 warehouse and 10 delivery staff randomly within the groups; total 60.
Example 5
Estimate a population count
Question
In a randomly selected sample of 100 pupils, 28 walk to school. The school has 750 pupils. Estimate how many walk and state a limitation.
Sample proportion = 28/100 = 0.28.
Estimate 210 pupils. This assumes the sample represents the school; another random sample could produce a different estimate.
Example 6
Large but biased
Question
A gym asks 2000 of its members whether they exercise weekly, to estimate the proportion of all adults who do. Explain why this may be worse for estimating all adults' behaviour than a smaller well-selected population sample.
Gym members are likely to exercise more than adults generally, so the selection excludes many less-active adults. The large size reduces random variation within that biased group, but does not make it representative of all adults. A smaller sample drawn from the target population can reduce that selection bias.
11 original questions · total 24 marks
Sampling GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 29 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
Population
You want to estimate the average commute of all staff in a hospital and ask 40 staff. What is the population?
Show worked answer
All staff in that hospital, not just the 40 surveyed.
Census
What distinguishes a census from a sample survey?
Show worked answer
A census attempts to collect data from every member of the population; a sample survey studies only a subset.
Random selection
Why is selecting the first 20 names on an alphabetic list not a simple random sample?
Show worked answer
Most names have no chance of being selected, while the first 20 are selected automatically. Use random distinct numbers across the whole list instead.
Allocation
A group has 60 boys and 90 girls. Allocate a proportional sample of 30.
Show worked answer
Total = 150.
Select 12 boys and 18 girls randomly within the groups.
One subgroup
A college has 800 students, including 240 in Year 12. How many Year 12 students belong in a proportional sample of 50?
Show worked answer
Allocate 15 places, then select randomly within Year 12.
Estimate
A well-selected sample of 80 people includes 18 who use a service. Estimate users in a population of 1200.
Show worked answer
Estimated users: 270. This is not an exact count of the population.
Leading question
Explain a problem with 'Don't you agree our excellent bus service should get more funding?'
Show worked answer
The wording encourages agreement by praising the service and suggesting the expected answer. Use neutral wording, such as 'Should bus-service funding increase, stay the same or decrease?'
Non-response
A random sample is invited to a survey, but only one quarter respond. Why could the results be biased?
Show worked answer
Respondents may differ systematically from nonrespondents, for example having stronger opinions. A random invitation does not guarantee a representative responding group.
Proportions and rounding
Groups contain 41 and 59 people. Allocate a proportional sample of 10 using nearest-whole-number allocations and check the total.
Show worked answer
The allocations 4 and 6 total 10. In other examples, independent rounding may not give the required total and would need adjustment.
A claim about certainty
A student says a random sample of 100 must perfectly match the population. Explain why this is wrong.
Show worked answer
Random selection gives fair selection chances, not guaranteed matching proportions in each sample. Chance variation remains. Larger well-selected samples tend to reduce it, but do not eliminate all uncertainty.
Capture–recapture estimate
Thirty animals are marked and released. A second random catch contains 45 animals, including 9 marked ones. Estimate the population and state an assumption.
Show worked answer
Let N be the population size. The second catch has marked fraction 9/45. Assume this approximates the fraction 30/N in the population.
About 150 animals. For example, assume the marked animals mix fully and have the same chance of recapture; the population must remain closed and marks must be retained.
Examiner-style feedback
Common sampling mistakes
How the group was selected matters as well as its size.
Randomly select within the allocated groups; do not simply ask the easiest people to reach.
A sample-based population total is an estimate and depends on representativeness.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
- Define the target population.
- Choose people without systematic exclusion.
- Preserve proportions when asked.
- Report uncertainty and possible bias.
Quick answers
Sampling FAQ
Is random the same as picking whoever is nearby?
No. Convenience selection gives some people little or no chance. Random selection uses a defined chance process over the intended population.
Is a proportional sample automatically unbiased?
No. The within-group selection, the population list, question wording and non-response can still introduce bias.
Content standards
Curriculum and rights review
S1/S5 inference and sampling limitations across tiers, with an explicitly labelled Edexcel Higher capture–recapture extension. Proportional subgroup allocation applies familiar ratio reasoning; this guide does not claim every exam board requires a separately named stratified-sampling algorithm. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
Official specification references