Sampling & Statistical Inference (Years 9–11) — Maths

Population vs sample, random/systematic/stratified sampling, sources of bias, and comparing data sets using both centre and spread.

Nobody surveys every single person in a country before an election, and nobody tests every light bulb off a factory line before selling any of them — that would be impossibly slow, expensive, or (for the light bulbs) would destroy the whole batch. Instead, statisticians ask a smaller group and use it to make a sensible judgement about the whole. That smaller group is a sample, and choosing it well is the difference between a trustworthy conclusion and a misleading one.

Why this matters: opinion polls, drug trials, quality control and school data all depend on sampling being done fairly. A badly chosen sample gave the wrong result in famous real elections and product recalls — examiners test whether you understand why a sampling method is fair, not just its name, and whether you can spot bias in how data was collected.

1. Population vs sample, and why we sample

The population is the entire group you are interested in — every student in a school, every widget from a factory, every voter in a country.
A sample is a smaller group taken from the population, used to draw conclusions about the whole population without checking every single member.
We sample because surveying or testing an entire population is usually too expensive, too slow, or (for destructive tests, like crash-testing a car) impossible to do on everything.

2. Sampling methods

Random sampling — every member of the population has an equal chance of being chosen, for example by assigning each person a number and using a random number generator to pick the sample.
Systematic sampling — choose every nth member from a list, after a random starting point. If a population of 500 needs a sample of 50, you would pick every 500/50 = 10th name.
Stratified sampling — split the population into groups (strata) that share a characteristic, such as year group or gender, then sample from each group in proportion to its size, so the sample mirrors the population's structure.

Stratified sampling: the sample's split across year groups mirrors the population's split.

Worked example: stratified sample size. A school has 500 students: 200 in Year 10, 125 in Year 11 and the rest in Year 12. A researcher wants a stratified sample of 50 students. How many should come from Year 10?
Proportion in Year 10 = 200/500.
Sample from Year 10 = (200/500) × 50 = 0.4 × 50 = 20 students.
The same method gives Year 11: (125/500) × 50 = 12.5, so 12 or 13 students, and the rest (17 or 18) from Year 12, keeping the sample proportional to each group's share of the school.

3. Bias in sampling

Bias happens when a sample systematically favours some members of the population over others, so it no longer represents the population fairly. Common sources of bias:

Only surveying people online — this misses anyone without internet access or who doesn't use that platform.
Only surveying people at one time of day or place, e.g. outside a train station at 8 a.m. only reaches commuters, not everyone.
Volunteer bias — people who choose to respond (e.g. to an optional online poll) often have stronger opinions than average.

Bias matters because a biased sample can lead to a confident-sounding but wrong conclusion about the whole population — the sample looks like solid evidence, but it was never capable of representing everyone in the first place.

4. Comparing two data sets: always centre AND spread

In s1 Averages & Range and s5 Histograms & Cumulative Frequency you found measures of centre (mean, median, mode) and measures of spread (range, interquartile range). Comparing two distributions properly means quoting one of each, because centre alone can hide a big difference in consistency.

Worked example: comparing two classes. Class A's test scores have mean 68 and range 10. Class B's test scores have mean 68 and range 40.
Both classes have the same mean — "on average" they did equally well.
But Class A's range is much smaller, so its scores are far more consistent — every student scored close to 68. Class B's wide range means some students did very well and others very badly.
Conclusion: the classes performed equally well on average, but Class A was more consistent (less spread out).

5. A simplified idea of confidence in a sample

A sample is only an estimate of the population — it won't match the population's true values exactly, and different samples give slightly different results. In general, the larger the sample, the more confident you can be that it reflects the population accurately, because a bigger sample averages out unusual individual results. This is why national opinion polls try to survey thousands of people rather than just a handful — a sample of 10 people could easily be unrepresentative by chance, while a sample of 2,000 is much less likely to be. (You do not need a formula for this at GCSE — just the idea that sample size affects how much you can trust the result.)

Common exam mistake: comparing two data sets using only an average ("Class A did better because its mean was higher") and never mentioning spread. A full comparison always needs both a measure of centre (mean/median) and a measure of spread (range/IQR) — a higher average with a huge spread is not simply "better".

Exam technique: when a question says "compare", write one sentence about the average (which is bigger/smaller) AND one sentence about the spread (which is more consistent) — examiners specifically award separate marks for each, so a comparison using only one measure can only ever get half the marks.