Population, census and sample

• Population — every member of the group being studied. Not necessarily people: it could be every light bulb from a factory, or every day of rainfall data.
• Census — data collected from the whole population. Completely accurate, but usually expensive, slow, and sometimes impossible.
• Sample — a subset of the population, used to estimate what the whole population is like.
• Sampling frame — a list of every member of the population (e.g. a school register), which you draw the sample from.

Why not always a census? Sometimes testing destroys the item. If you crash-test every car you produce, you have no cars left to sell. Sampling is not a compromise here — it is the only option.

Bias is when a sample systematically misrepresents the population. A bigger sample does not fix bias — it just gives you a more precise wrong answer.

Census or sample?

Random sampling methods

In a random method, chance decides who is chosen — so it can be free from human bias.

• Simple random sampling — every member of the population has an equal chance of being chosen. Number the sampling frame, then use random numbers. Fair and unbiased, but needs a complete sampling frame and can be slow for large populations.
• Systematic sampling — order the frame, choose a random start, then take every kth member, where k = N/n. Quick and simple, but can go badly wrong if the list has a repeating pattern with the same period as k.
• Stratified sampling — split the population into strata (groups), then sample each stratum in proportion to its size. Best representation, but you must know the strata sizes in advance.

Stratified sampling — worked

When numbers do not come out whole, round sensibly and adjust so the parts still add to the required total. Say what you did.

Your turn — stratified sample

Your turn — the other stratum

Systematic sampling — worked

The danger: if the list has a hidden periodic pattern matching k, the sample is badly biased. Sampling every 7th day of a rota, for example, could pick only Mondays — and Mondays may be nothing like the rest of the week.

Your turn — the sampling interval

Your turn — which member?

Non-random methods

• Opportunity (convenience) sampling — take whoever is available. Very quick and cheap, but highly likely to be biased — the people who happen to be there are usually not typical.
• Quota sampling — the interviewer is told to fill fixed quotas (e.g. 20 men, 20 women), choosing subjects themselves. No sampling frame needed, but the interviewer’s choices introduce bias.
• Cluster sampling — split the population into naturally occurring clusters and randomly select whole clusters. Cheap for spread-out populations, but a cluster may not represent the whole.

Exam phrasing: when asked to “comment on the sampling method”, name one advantage and one disadvantage in context. “Surveying shoppers on a Tuesday morning is opportunity sampling — quick and cheap, but people at work are excluded, so the sample is not representative of all shoppers.”

Spot the bias

Your turn — a fair chance

The OCR pre-release large data set

OCR issues a pre-release large data set (its name for the large data set) that you are expected to have worked with during the course. Questions in the exam may be set in its context.

• You will not have a printout of the pre-release data set in the exam.
• Any values you actually need for a calculation will be given to you in the question.
• What is assessed is your familiarity with it: what its variables mean, what units they use, how the data was collected and sampled, and what its limitations are.

How to prepare properly: get the current data set from your teacher or the OCR website — it is specific to your exam series. Explore it in a spreadsheet: find the outliers, notice which variables have missing entries, and ask yourself which sampling method produced it and what bias that might introduce. That is exactly the thinking the exam rewards.

The data set in the exam

Define the sampling frame

Name that sampling method

Tap a description, then tap the sampling method it describes.

Statistical vocabulary

Tap a term on the left, then its definition.

The big ideas to know

That is OCR 2.01 — the foundation every other statistics question stands on. Press Finish to see your score.