Populations, samples and the large data set

What a population, a census and a sample actually are; the sampling units and the sampling frame; why you would sample rather than take a census (and when a census is impossible); and how a sample can be unrepresentative — including the trap of combining overlapping samples.

Statistics · Statistical sampling

Populations, samples and the large data set

Why it works

Statistics starts with a simple split: the population is the whole set of things you care about — every member of it — and a sample is a subset you actually look at. If you were studying the heights of all Year 12 students in a school, the population is every Year 12 student there; a sample might be 30 of them.

A census observes every member of the population. It is completely accurate in principle — you've missed no one — but it is often expensive, slow, or simply impossible:

Cost and time. Asking all 30 million voters, or measuring every tree in a

forest, is huge.

Destructive testing. To find how long a make of lightbulb lasts you have to

run bulbs until they fail. A census would destroy the entire stock — there'd be nothing left to sell. Here you must sample.

A sample is cheaper and faster, and lets you study a population that's too big (or too destructible) to examine fully. The price you pay is that a sample might not perfectly reflect the population — that's sampling error — and a badly chosen one can be biased.

The vocabulary that earns the marks. Examiners are fussy about three words:

Sampling units — the individual members of the population that can be

sampled (one student, one bulb, one day's weather reading).

Sampling frame — a list of all the sampling units, often named or

numbered (a register, a membership list). You need one to take a random sample, and not every population has one (there's no list of "all the fish in a lake").

Population — all the units; never use it to mean the sample.

Census vs sample, at a glance. A census counts every unit, so it is unbiased in principle but costs the most in time, money and data handling, and is impossible when testing is destructive or the population is enormous. A sample sees only part of the population, so it is subject to sampling error, but it is cheaper, quicker, and usually the only feasible option.

Representative, not just big. A good sample represents the population, so conclusions about the sample carry over to the population. Bias comes from the method, not the size: a survey of phone-box users, or only your friends, is biased no matter how many people you ask. Making a biased sample larger just gives you a more confident wrong answer.

A subtler trap: overlapping samples can't simply be combined. If one student samples a data set "from the 1st, every 10th reading" and another samples it "from the 5th, every 10th reading", and you then pool the two, some of the same readings may be counted twice (or the two cover overlapping stretches). The pooled results are then unreliable, because the sampling units are no longer distinct.

The large data set. Edexcel issues a large real-world data set (weather records from several UK and overseas stations across given months) that you're expected to have explored before the exam. You won't reproduce it from memory; the exam tests whether you can sample from it, spot what type each variable is, and interpret values in context — for example knowing that some locations and months tend to be warmer, wetter or windier. Throughout Figure we teach those same skills on our own weather-style data, so the method is identical even though the numbers differ.