This mini-lesson covers AQA section K โ Statistical sampling: the terms population and sample, census vs sample, sampling frames, and the five techniques you must be able to describe and critique โ simple random, systematic, stratified, quota and opportunity sampling.
A statistic from a sample estimates a parameter of the population โ different samples give different answers.
Work through each screen, answer the questions as you go (some are wordy, some are calculations) and collect โญ stars. AQA's spec names simple random and opportunity sampling explicitly, and expects you to select or critique any technique in context. Press Start when you're ready.
Section K1 ยท language
Population, sample, census
A population is every member of the group you want to know about. A sample is a subset of it that you actually measure.
A census collects data from the whole population: totally accurate in principle, but expensive, slow, and often impossible (e.g. testing every match in a factory destroys the stock).
A sample is quicker and cheaper, but it may not be representative, so conclusions carry uncertainty.
A sampling frame is a list of all the members of the population (e.g. a school register). Random methods need one.
AQA insists on this: different samples from the same population can lead to different conclusions. A sample statistic (e.g. the sample mean xฬ) is only an estimate of the population parameter (ฮผ).
Quick check
Census or sample?
?A tyre factory wants to know the mean lifetime of its tyres. Testing a tyre destroys it. Should they take a census or a sample, and why?
Section K1 ยท methods
The five sampling techniques
You must be able to describe, use and critique each of these.
Simple random sampling โ number every member of the sampling frame, then use random numbers (or a calculator's RAN# key) to pick n of them. Every sample of size n is equally likely. Free of bias, but needs a full frame and can be slow for a large population.
Systematic sampling โ order the frame, choose a random start between 1 and k, then take every k-th member, where k = N รท n. Quick and simple, but if the list has a periodic pattern matching k, the sample is badly biased.
Stratified sampling โ split the population into strata (e.g. year groups) and sample each stratum in proportion to its size, sampling randomly within each stratum. Reflects the population structure; but you must know the strata sizes, and it is more work.
Quota sampling โ the interviewer is told to fill fixed quotas (e.g. 20 men, 20 women) using whoever they meet. Non-random, so it can be badly biased by the interviewer's choices, but it is cheap and no frame is needed.
Opportunity (convenience) sampling โ take whoever is available at the time. Easiest and cheapest of all, and the least likely to be representative.
systematic interval k = N รท nN = population size ยท n = sample size ยท start at a random number from 1 to k
Calculate
Stratified sample size
1A school has 1200 students, of whom 450 are in Year 12. A stratified sample of 80 students is taken. How many should come from Year 12?
students
Hint: (450 รท 1200) ร 80 = 0.375 ร 80.
Calculate
The sampling interval
2A factory produces 900 components in a shift. A quality inspector wants a systematic sample of 60. Calculate the sampling interval k.
Hint: k = N รท n = 900 รท 60.
Calculate
Systematic: which member?
3A population of 800 is listed and a systematic sample of 50 is taken (so k = 16). The random start is member number 7. Which member is selected third?
Tap a statement, then tap the sampling method it describes.
๐ฒ Simple random
๐ข Systematic
๐ถ Opportunity
Section K1 ยท critique
Bias โ and why samples disagree
A sampling method is biased if some members of the population are systematically more likely to be chosen than others.
Non-response and self-selection (only keen people reply) bias a survey even if the selection was random.
An opportunity sample outside a gym at 7am tells you about early-morning gym-goers โ not about the town.
Quota sampling is non-random: the interviewer may avoid people who look unfriendly.
Worked example โ estimating from a sample
A random sample of 8 daily rainfall readings (mm) is: 12, 15, 9, 14, 11, 13, 10, 12.
ฮฃx = 96, so xฬ = 96 รท 8 = 12 mm. This is an estimate of the population mean ฮผ.
A different random sample of 8 days would almost certainly give a different xฬ โ that variability is exactly why we need hypothesis tests (section O).
Exam phrasing: "Give one advantage and one disadvantage of this method in this context." Always tie the answer to the context โ generic answers score zero.
Calculate
Estimating the mean
4A random sample of 8 daily rainfall readings (mm) is 12, 15, 9, 14, 11, 13, 10, 12. Calculate the sample mean xฬ.
mm
Hint: Add them: ฮฃx = 96. Then xฬ = ฮฃx รท n = 96 รท 8.
Calculate
Sampling fraction
5A college has 900 students and a sample of 45 is taken. Calculate the sampling fraction n รท N as a percentage.
%
Hint: 45 รท 900 = 0.05, then ร 100.
Quick check
Name that method
?A researcher stands outside a supermarket at 11am on a Tuesday and interviews the first 50 people who walk past. Which technique is this?
Quick check
Quota or stratified?
?An interviewer is told to question 20 men and 20 women, choosing whoever they meet until each group is full. Which statement is correct?
Match it
Key terms
Tap a description on the left, then its term on the right.
Description
Term
Quick check
Critique the method
?What is the main disadvantage of systematic sampling?
Section K1 ยท in practice
Taking a simple random sample properly
"Choose 20 people at random" is not a method. Examiners want the mechanism:
Number every member of the sampling frame 1 to N (e.g. 001โ500).
Generate random numbers (calculator RAN#, random number table, or a computer).
Take the members with those numbers, ignoring repeats, until you have n.
The lottery method (all names in a hat, drawn without replacement) is the same idea.
Non-response and self-selection: even a perfect random sample is spoiled if the people you select refuse to answer, or if only people with strong opinions reply. Both introduce bias that a bigger sample will not fix.
Quick check
Name that method
?All 500 members of a club have their names written on identical slips, which are shuffled in a box. Twenty slips are drawn out without replacement. Which sampling method is this?
Calculate
Stratified sample again
6A college has 1500 students, of whom 500 are in Year 13. A stratified sample of 60 is taken. How many students should come from Year 13?
students
Hint: (500 รท 1500) ร 60 = (1/3) ร 60.
Recap
The big ideas to know
Population vs sample: a sample statistic (xฬ) estimates a population parameter (ฮผ)
Census: everyone โ accurate but costly, slow, sometimes destructive
Sampling frame: a list of the whole population โ needed for random methods
Simple random: equal chance for every sample ยท Systematic: every k-th, k = N รท n, random start
Stratified: each stratum in proportion ยท Quota & Opportunity: non-random, cheap, risk of bias
Always: different samples โ different conclusions; critique the method in context
That is the whole of section K โ Statistical sampling for AQA A-level Mathematics (7357). Press Finish to see your score.
๐
Mini-lesson complete!
โญโญโญ
You've worked through Statistical sampling for AQA A-level Mathematics. ๐
Your stars: 0 / 0
Next: test yourself in the Evaluate stage Confidence Quiz, then lock it in with Verify.
๐ฃ Smashed it? Share your score
Challenge a mate to beat your stars, or show a parent how you got on.