This mini-lesson covers populations vs samples, the census, sampling frames, the five sampling methods you must be able to describe and criticise (simple random, systematic, stratified, quota, opportunity) and how bias creeps in.
Where this sits:Unit AS 2: Applied Mathematics — statistical sampling: sampling terminology, methods of sampling, and selecting a sample from a population, including work with a large data set.
Work through each screen, answer the questions as you go (some are reasoning, some are calculations) and collect ⭐ stars. Press Start when you are ready.
Terminology
Population, census and sample
Population — every member of the group you are studying.
Census — data from every member. Completely accurate, but expensive, slow, and sometimes impossible (e.g. testing every match in a factory would destroy the stock).
Sample — a subset used to estimate facts about the population. Cheap and quick, but it may not represent the population.
Sampling frame — the list of population members (e.g. a school roll) from which you sample. Without a frame you cannot take a simple random sample.
Bigger samples give more reliable estimates but cost more. A badly chosen large sample is worse than a well-chosen small one — size does not cure bias.
Random methods
Simple random, systematic, stratified
Simple random: every member has an equal chance of selection — number the frame and use random numbers. Unbiased, but needs a full frame and may by chance miss a subgroup.
Systematic: pick every kth member after a random start, where k = population ÷ sample size. Quick and spread out — but biased if the list has a hidden pattern of period k.
Stratified: split the population into strata (groups), then sample each stratum in proportion to its size, randomly within it. Reflects the population structure — but you must know the strata sizes.
stratum sample = (stratum size ÷ population) × sample sizeround sensibly, and check the parts add back to the total
Calculate
Stratified sample
1A school has 800 pupils: 500 girls and 300 boys. A stratified sample of 80 pupils is taken. How many girls should be in the sample?
girls
Hint: Girls make up 500/800 of the school, so they get (500 ÷ 800) × 80 = 0.625 × 80 of the sample.
Calculate
Systematic interval
2A systematic sample of 40 is taken from a population of 1200. What is the sampling interval k?
Hint: k = population ÷ sample size = 1200 ÷ 40. (You then choose a random start between 1 and 30 and take every 30th member.)
Calculate
Three strata
3A college has 240 Year 12 students, 180 Year 13 students and 60 staff. A stratified sample of 40 is taken. How many Year 13 students should be sampled?
students
Hint: Total = 240 + 180 + 60 = 480. Year 13 share = 180/480 = 0.375, so 0.375 × 40 = 15.
Quick check
Spot the method
?A researcher stands at a shopping-centre door and interviews the first 50 people who walk past. Which method is this?
Non-random methods
Quota and opportunity sampling
Quota sampling: the interviewer is told to fill fixed numbers in each group (e.g. 20 men, 20 women) but chooses who to ask. No sampling frame needed — but the interviewer's choice introduces bias, and it is not random.
Opportunity (convenience) sampling: use whoever is available. Fastest and cheapest, but the least representative and it depends heavily on the individual doing the sampling.
Exam wording: if a question asks for an advantage and a disadvantage, give one of each in context — "quick and cheap" alone is not enough; say why it may not represent the pupils in this school.
Calculate
Sampling fraction
4A sample of 45 is taken from a population of 900. Express the sampling fraction as a percentage.
%
Hint: 45 ÷ 900 = 0.05, and 0.05 × 100 = 5%.
Calculate
Equal chance
5In a simple random sample of size 1 from a club of 250 members, what is the probability that a particular named member is chosen? Give your answer as a decimal.
Hint: Every member is equally likely, so the probability is 1 ÷ 250 = 0.004.
Sort it
Random or not?
Tap a method, then tap the family it belongs to.
🎲 Random method
🙋 Non-random method
👥 Whole population
Quick check
Why stratify?
?A school is 60% girls. A simple random sample of 20 pupils happens to contain 18 boys. What does stratified sampling fix?
Match it
Match the sampling term
Tap the term on the left, then its definition on the right.
Statement
Answer
Quick check
Systematic danger
?A factory line makes items in a repeating cycle of 10, where every 10th item comes from a faulty machine. A systematic sample takes every 10th item. What goes wrong?
Quick check
Census or sample?
?A firm tests the lifetime of its light bulbs by running each one until it fails. Should it use a census?
Examiner traps
Sampling pitfalls
Vague advantages. "Quick and cheap" alone earns nothing — say why it matters in this context.
Systematic ≠ random start ignored. A systematic sample needs a random starting point within the first k.
Stratified sizes must be proportional, and the parts must add back to the total sample size.
Bias is not cured by size. A bigger biased sample is still biased.
Simple random sampling needs a sampling frame — without a full list you cannot do it.
Recap
The big ideas to know
Census = everyone (accurate, costly, sometimes destructive) · Sample = a subset (quick, may be unrepresentative)
Sampling frame = the list you sample from — needed for simple random sampling
Random: simple random · systematic (k = N/n, random start) · stratified (proportional to strata)
Non-random: quota (interviewer fills groups) · opportunity (whoever is available)
Bias: a bigger sample does not fix a biased method
Good sampling is what makes every later inference — probability, distributions, hypothesis tests — trustworthy. Press Finish to see your score.
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