Maths › Statistics › Sampling and the large data set
Sampling and the large data set
You can rarely measure everyone, so you measure some and argue about the rest. This lesson is about when that argument holds up: what each sampling method buys you, what it costs, and what the exam's own data set looks like.
IN THIS TOPIC
- Define population, census, sample and sampling frame, and say when a census is impractical.
- Describe simple random, systematic, stratified, quota and opportunity sampling, with one advantage and one limitation of each.
- Know the shape of the large data set: the weather variables, the stations and the periods.
WHAT YOU PROBABLY THINK
A larger sample automatically makes the results representative.
Why sample at all
A population is every member of the group you care about; a census measures all of them. Sometimes that is fine: a teacher can ask every student in one class. But a census of a large population is slow and expensive, and sometimes impossible on principle, because the measuring destroys the thing measured. You cannot crash-test every car.
So you take a sample: a subset, measured fully, standing in for the rest. The whole subject of statistics is the argument from sample to population, and the argument is only as good as the way the sample was chosen. The list you choose from, every population member with an identifier, is the sampling frame.
The random methods
A simple random sample of size n gives every possible group of n an equal chance: number the frame, then use a random number generator or lottery draw. It is the gold standard for fairness, and the standard against which everything else is judged. Its cost is practical: you need a full frame, and the chosen members may be scattered and hard to reach.
Systematic sampling takes every kth member of the frame after a random start: for 50 from 1000, k = 20, start anywhere in the first 20. Quick to run on a long list, and still random, provided the list has no repeating pattern that lines up with k.
Stratified sampling splits the population into groups that matter, strata, and takes a simple random sample from each, sized in proportion. The number from each stratum is
so a school of 240 sampled 30 strong takes one person in eight from every stratum. The sample inherits the population's structure by construction; the price is needing to know the strata and their sizes in advance.
The non-random methods
Quota sampling keeps the proportions but drops the randomness: an interviewer fills quotas, so many from each group, choosing whoever comes to hand until each quota is full. No frame needed, fast, cheap, and the standard tool of street surveys. But the interviewer's choices can smuggle in bias that no arithmetic can remove.
Opportunity sampling (or convenience sampling) simply takes whoever is available: the first twenty people through the door. It is the quickest method and the weakest, because availability is rarely unrelated to what you are measuring. Ask about exercise habits at a gym entrance and the sample answers a different question.
WORKED EXAMPLE
Choosing and using a method
A sixth form has 600 Year 12 and 400 Year 13 students. Explain how to take a stratified sample of 40 by year group.
The fractions come first: Year 12 supplies 40 × 600/1000 = 24 students and Year 13 supplies 40 × 400/1000 = 16.
Then each group of names is numbered and a simple random sample of the right size is drawn from each, using random numbers. Both halves of the answer earn marks: the proportional arithmetic, and the random selection within each stratum.
The large data set
Edexcel's large data set is real weather-station data, and the paper assumes you have worked with it. It records daily values, temperature, rainfall, sunshine, wind speed and direction, cloud cover, visibility and pressure, for the months May to October in two years, 1987 and 2015.
The stations matter: five in the UK, Camborne, Heathrow, Hurn, Leeming and Leuchars, and three overseas, Beijing, Jacksonville and Perth. Perth is in the southern hemisphere, so its May-to-October window is winter, a favourite exam trap. Rainfall entries of tr mean a trace, less than 0.05 mm, and some entries are missing: real data needs cleaning before it needs summarising.
THE EXAM BIT
- Name the method a description matches, then give an advantage and a limitation in context, not from a memorised list: the marks are for this sample, this population.
- Stratified calculations are two marks of arithmetic: stratum over population, times sample size.
- For the large data set, know the stations, the two years, the May-to-October window, and that Perth's seasons run opposite to the UK's.
- “Random” has a technical meaning: every member (or every group of n) equally likely. Quota and opportunity sampling are not random, whatever their proportions.
CHECK YOURSELF
A factory's 1200 workers are 900 full-time and 300 part-time. Describe how to take a sample of 60, stratified by contract type.
Show a hint
Fractions of the population first, then random selection inside each stratum.
Show the answer
Full-time supplies 60 × 900/1200 = 45 workers; part-time supplies 60 × 300/1200 = 15.
Number each group's members and draw a simple random sample of the required size from each, for example with a random number generator.
A sample speaks for its population only if chance, not convenience, chose it.
Stratify when you know the structure: the same fraction from every stratum keeps the sample a scale model.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Define population, census, sample and sampling frame, and say when a census is impractical.
- Describe simple random, systematic, stratified, quota and opportunity sampling, with one advantage and one limitation of each.
- Know the shape of the large data set: the weather variables, the stations and the periods.
No animated video for this topic yet; these notes stand alone.