Sampling and bias
Maths · GCSE · The School
Population and sample
The population is everyone or everything the question is about — every pupil in a school, every bolt in a batch, every household in a town. A census asks all of them. Usually that is too slow, too expensive, or destroys the thing being tested: you cannot test every bolt to destruction and still have bolts to sell. So we take a sample and use it to say something about the population. The whole subject turns on one idea: a sample is only useful if it resembles the population it came from, and everything below is about making that likely.
Random, systematic and stratified
A simple random sample gives every member of the population an equal chance of being chosen — number them all and pick numbers at random. Systematic sampling takes every nth one from a list after a random start, which is quicker but goes wrong if the list has a repeating pattern of the same period. Stratified sampling splits the population into groups that matter, then takes from each group in proportion to its size. If a school of 1000 has 600 girls and 400 boys and you want a sample of 50, you take 30 girls and 20 boys, because 600/1000 of 50 is 30.
How samples go wrong
Bias means the sample systematically over-represents part of the population, and a bigger sample does not fix it — it just makes you more confident about the wrong answer. Asking about sport in the queue for a football match is convenience sampling. Putting a survey online and letting people opt in is self-selection, which reaches the people who feel strongly. Asking "do you agree that homework is excessive?" leads the answer. Surveying at 10am on a weekday misses everyone at work. In each case the fault is in the METHOD, not the arithmetic that follows it.
A town of 8000 people has 2000 under 18, 4500 aged 18 to 64 and 1500 aged 65 and over. You want a stratified sample of 400. Each stratum contributes its own share: 2000/8000 x 400 = 100 under 18; 4500/8000 x 400 = 225 aged 18 to 64; 1500/8000 x 400 = 75 aged 65 and over. Those add to 400, and checking that they do is the fastest way to catch an arithmetic slip. Then choose randomly WITHIN each group — stratifying decides how many come from each stratum, not which individuals, and picking the convenient ones inside a stratum puts the bias straight back in.