Every statistical question starts with a choice: study everyone, or study a few and generalise. That choice — population versus sample — shapes every method that follows.
8 min read4 worked examples
Population, census and sample
A population is the entire set of items or individuals that a statistical investigation is interested in — every student at a school, every component off a factory line, every day of weather at a location. A census collects data from every member of the population. A sample collects data from only some members, chosen to represent the whole.
Why sample instead of taking a census?
A census is the only way to get data with no sampling error at all — but it's often impossible in practice. It can be too expensive and too time-consuming for a large population, and some testing is destructive (crash-testing every car off a production line would leave none to sell). A well-chosen sample trades a small, measurable amount of uncertainty for a large saving in cost and time.
The right choice depends on the population's size and the cost of measuring each member. A school register of 800 pupils can realistically be censused; the lifespan of every lightbulb a factory produces cannot.
Sampling frames and sampling units
To take a sample, you first need a sampling frame — a list of every member of the population, each one individually identifiable. A single member of that list is a sampling unit. The sample itself is then a subset of units drawn from the frame.
A frame is only ever an approximation of the population
A perfect sampling frame lists every population member exactly once, with no omissions and no duplicates — but real frames rarely manage this. The electoral register misses people who haven't registered to vote; a school register misses pupils who joined after it was printed. Any mismatch between the frame and the true population is a source of bias before a single sample is even drawn.
Worked example 1
Identify population, frame and unit
A researcher wants to investigate the average commute time of employees at a company with 3,000 staff. She uses the company's HR database, which lists all 3,000 employees, and selects 150 of them to survey. Identify the population, the sampling frame, and a sampling unit.
MethodMatch each term to its role in the scenario
1
The population is every employee the study is about: all 3,000 staff at the company.
The full group of interest.
2
The sampling frame is the HR database — the actual list used to select from.
The list, not the group itself.
3
A sampling unit is a single employee on that list.
One listed, selectable individual.
▪ identified
Examiner's eye
Population, frame and unit are commonly confused. The population is the group you care about; the frame is the list you actually have access to; a unit is a single entry on that list. Precise wording earns the mark — "the employees" alone is too vague for "sampling frame".
Worked example 2
Census or sample?
For each situation, state whether a census or a sample is more appropriate, with a reason: (a) testing the strength of matches in a box of 40; (b) checking the ages of all 25 students in one class before a school trip.
MethodWeigh population size, cost, and whether testing destroys the item
1
(a) A sample. Testing a match's strength means burning or breaking it — a destructive test. A census would destroy every match in the box, leaving none to sell.
Destructive testing rules out a census.
2
(b) A census. The population (25 students) is small, easily accessible, and checking ages is quick and non-destructive — there's no real cost saving from sampling.
Small population, cheap to check, no reason to sample.
▪ decided
The key insight
There's no single "correct" answer independent of context — the decision always weighs population size against cost, time, and whether measuring destroys the item. A census is preferred whenever it's practical; a sample becomes necessary once a census would be too costly, too slow, or physically impossible.
Worked example 3
Spotting a flawed sampling frame
A council wants to survey the opinions of all residents of a town on a new park. They use the town's landline telephone directory as their sampling frame. Explain one way this frame might not accurately represent the population of residents.
The population is "all residents of the town" — but the frame (the phone directory) is a list of landline numbers, not a list of residents.
MethodCompare who's on the frame against who's in the population
1
Residents without a landline (many younger people rely on mobiles only) don't appear in the directory at all.
An omission from the frame.
2
A household with a landline may have several residents, but the directory lists only one entry per household.
Some residents are effectively invisible even if their household is listed.
3
So the frame under-represents younger residents and residents who aren't the named account holder — the sample drawn from it would be biased toward older, landline-owning residents.
The mismatch translates directly into sampling bias.
▪ explained
Common mistake
"The frame is out of date" is a common but often lazy answer — better answers name a specific group the frame misses or under-represents (here, non-landline residents and unlisted household members) and explain why that skews the sample's opinions away from the true population's.
Worked example 4
Designing a sampling frame · Challenge
A university wants to sample its 12,000 students to gauge satisfaction with campus food outlets. Suggest a suitable sampling frame, and explain one limitation it might still have.
MethodChoose an existing, complete list, then interrogate it for gaps
1
A suitable frame: the university's student enrolment database, which should list every currently registered student exactly once.
A limitation: students who rarely visit campus (e.g. those studying fully online, or on placement year) are on the frame but have little or no experience of the food outlets being asked about.
Being on the list doesn't guarantee relevant experience.
3
Their responses (or non-responses) could distort the results even though the frame itself is technically complete and accurate.
A "correct" frame can still produce a misleading sample.
▪ discussed
Look deeper — completeness isn't the same as suitability
A sampling frame can be perfectly accurate and up to date, and still be a poor choice if it includes members who can't meaningfully answer the question being asked. Choosing a frame is not just "find a complete list" — it's "find a complete list of people the question actually applies to".
Common mistake
Don't confuse "sample size" with "sampling frame" — the frame is the list a sample is drawn from, and the sample is the subset actually selected. And a census is not automatically "better" than a sample: for a huge or destructible population, a well-designed sample can be the only realistic option.
Check yourself
A quality inspector wants to check the diameter of bolts produced by a machine that makes 50,000 bolts a day. State whether a census or sample is more appropriate, and give a reason. (Hint: think about how many bolts there are, and how long checking each one would take.)
Method
1
A sample is more appropriate.
2
Checking all 50,000 bolts every day would be far too time-consuming and costly to be practical.
Large population, high measuring cost.
3
A sample gives a good estimate of the machine's accuracy at a fraction of the cost, and can be repeated regularly through the day.