This mini-lesson covers Topic 1 of Edexcel A-level Mathematics (9MA0): populations, censuses and samples, the five sampling methods you must know — simple random, systematic, stratified, quota and opportunity (convenience) — and how to select or critique a method for a given context, including the large data set.
A census surveys the whole population; a sample surveys part of it and is used to make an inference about the population.
Work through each screen, answer the questions as you go (some are wordy, some are calculations) and collect ⭐ stars. Throughout, take g = 9.8 m s−2 unless a question says otherwise. Press Start when you are ready.
Sampling · the language
Population, census, sample
A population is the whole set of items you are interested in (e.g. every student in a school).
A census observes or measures every member of the population.
A sample is a selected subset. The individual units are the sampling units; a numbered list of all of them is the sampling frame.
Census vs sample — the trade-off
Census: ✔ completely accurate, every member counted. ✘ time-consuming, expensive, and often impossible (testing every match in a factory would destroy them all).
Sample: ✔ quicker and cheaper, less data to process. ✘ the data may not be representative, and the sample may introduce bias.
Bias creeps in when some members of the population are more likely to be chosen than others, or when a group is left out of the sampling frame altogether. Any conclusion is then unsafe — no matter how big the sample.
Quick check
Naming the parts
?A researcher writes down a numbered list of all 480 employees at a firm and then chooses from it. What is this list called?
Sampling · random methods
Simple random, systematic & stratified
These three are random methods: chance decides who is picked, so bias is reduced.
Simple random sampling — every sample of size n has an equal chance of being selected. Use random numbers, or draw names from a hat. ✔ free of bias, easy and cheap for small populations. ✘ needs a full sampling frame; not suitable for a large population.
Systematic sampling — order the population, then take every k-th member, where k = population size ÷ sample size. The starting point in the first k must be chosen at random. ✔ simple, quick, suitable for large populations. ✘ a periodic pattern in the list can introduce bias.
Stratified sampling — split the population into strata (groups), then sample each stratum in proportion to its size, using simple random sampling within each: number from a stratum = (stratum size ÷ population size) × sample size. ✔ reflects the population structure, so it is likely to be representative. ✘ the population must divide into clear, non-overlapping groups.
stratum sample = (stratum size ÷ population size) × nsystematic interval k = population size ÷ sample size
Calculate
Stratified sample
1A sixth form has 1200 students: 750 in Year 12 and 450 in Year 13. A stratified sample of size 80 is taken. How many students are sampled from Year 12?
students
Hint: (750 ÷ 1200) × 80 = 0.625 × 80.
Calculate
Systematic interval
2The same sixth form of 1200 students is sampled systematically to give a sample of size 80. Find the sampling interval k.
Hint: k = population ÷ sample size = 1200 ÷ 80.
Sampling · non-random methods
Quota & opportunity sampling
These two are non-random: a person, not chance, decides who ends up in the sample. They are quick, but they can be badly biased.
Quota sampling — the population is split into groups by characteristics (age, sex, ...). The interviewer is told how many to get from each group (their quota) and then chooses people until the quota is filled. ✔ small samples possible, costs are low, and no sampling frame is needed. ✘ not random, so it can be biased; non-responses are not recorded; the groups must be chosen carefully.
Opportunity (convenience) sampling — take whoever is available at the time and fits the criteria (e.g. the first 20 people leaving a shop). ✔ easy, cheap, instant. ✘ unlikely to be representative; heavily dependent on the researcher, so it can be very biased.
Exam skill (spec 1.2): you must select or critique a method. Always link the criticism to the context: "surveying shoppers on a Tuesday morning is opportunity sampling, so people who work weekdays are excluded and the sample is unlikely to be representative of all shoppers."
Sort it
Which sampling method?
Tap a description, then tap the method it defines.
🎲 Simple random
🔢 Systematic
🧩 Stratified
Quick check
Name that method
?A market researcher stands in a high street and interviews people until she has spoken to 30 men and 30 women. Which sampling method is this?
Calculate
Stratified again
3A factory runs three shifts: shift A has 250 workers, shift B has 150 and shift C has 100. A stratified sample of 60 workers is taken. How many come from shift B?
workers
Hint: Population = 250 + 150 + 100 = 500. So (150 ÷ 500) × 60.
Sampling · carrying it out
Doing it properly: random starts & rounding
Systematic sampling — worked example
A population of 1000 people is listed and a sample of 25 is required.
k = 1000 ÷ 25 = 40. Choose a random start between 1 and 40 — say 7.
The sample is then person 7, 47, 87, 127, ... (add 40 each time). The 3rd person chosen is 7 + 2 × 40 = 87.
Stratified sampling — rounding
A firm has 250 staff; 63 work in Sales. A stratified sample of 30 is taken.
Sales: (63 ÷ 250) × 30 = 7.56 → round to the nearest whole number, 8 people.
Do this for every stratum, then check the totals add to the required sample size (adjust by one if rounding pushes you over).
Large data set: Edexcel's data set is a sample of UK/overseas weather data. Rows with "n/a" mean no data was recorded, not zero — you must clean the data before you calculate anything, and a location's readings are only a sample of that location's weather.
Calculate
Systematic in practice
4A population of 1000 is sampled systematically to give 25 people. The random starting number is 7. Which member of the list is the 3rd person selected?
th person
Hint: k = 1000 ÷ 25 = 40. The people chosen are 7, 7 + 40, 7 + 2 × 40, ...
Calculate
Stratified with rounding
5A firm has 250 staff, of whom 63 work in Sales. A stratified sample of 30 staff is taken. How many Sales staff should be in the sample (to the nearest whole number)?
people
Hint: (63 ÷ 250) × 30 = 7.56, then round.
Match it
Method → key point
Tap an item on the left, then its partner on the right.
Statement
Method
Quick check
Census: the catch
?Which of these is a genuine disadvantage of a census compared with a sample?
Quick check
Critique the method
?A researcher samples every 10th house on a street, starting from a randomly chosen house among the first 10. Every 10th house happens to be a corner house with a large garden. What is the problem?
Sampling · the large data set
The large data set
Edexcel's large data set is real weather data from five UK and three overseas locations. It is itself a sample, so everything in this topic applies to it.
Each location's rows are a sample of days — the readings from May to October only, so you cannot infer anything about winter.
Some cells contain "n/a". That means no reading was taken, not zero. Exclude those days from a calculation and say that you have.
"tr" (trace) rainfall means less than 0.05 mm — it is usually taken as 0 for calculations, but you must state the assumption.
Typical question: "Explain why the mean daily rainfall for Camborne calculated from the large data set may not be a good estimate of the mean daily rainfall for the whole year." Answer: the data covers only May to October, so it is not a random sample of the whole year and will under-represent the wetter winter months.
Quick check
Cleaning the large data set
?In the large data set, some daily rainfall entries are recorded as n/a. What should you do with those days when calculating a mean?
Recap
The big ideas to know
Census: every member — totally accurate, but slow, costly, sometimes destructive
Sample: a subset — quick and cheap, but risks bias and non-representativeness
Simple random: every sample of size n equally likely; needs a full sampling frame
Systematic: every k-th item, k = population ÷ sample size, with a random start in the first k
Stratified: stratum sample = (stratum size ÷ population) × n — most likely to be representative
Quota: non-random, interviewer fills a target per group · Opportunity: whoever is available
Always justify a choice, and criticise it in context (bias, missing groups, non-response)
That is the whole of 9MA0 Topic 1 — Statistical sampling. Press Finish to see your score.
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