Population, sample and sampling techniquesEdexcel A-Level Maths: Revision notes

The population is the whole set of items or people you want information about. A sample is a selection from it. A census observes every member of the population. We use a sample to make informal inferences about the population, such as estimating a mean or a proportion. The list of all members from which a sample is drawn is the sampling frame.

• Census: gives the true value and is unbiased in selection, but is expensive, slow and sometimes impossible (for example if testing destroys the item, such as crash-testing cars).
• Sample: cheaper, quicker and less data to process, but only an estimate and may not be representative.

In a simple random sample of size $n$, every possible sample of size $n$ has an equal chance of being selected, so every member has an equal chance. Method: number each member of the sampling frame $1$ to $N$; generate $n$ random numbers in that range with a calculator or random number generator; ignore repeats; select the members with those numbers.

• Advantages: free from researcher bias and each member has an equal chance.
• Disadvantages: needs a complete sampling frame, and may be impractical if the population is large or spread out. A sample can by chance miss a small group.

Systematic sampling: choose a random start, then take every $k$th member of an ordered list, where $k=\frac{N}{n}$. Quick and simple, but biased if the list has a pattern that matches the interval. Stratified sampling: divide the population into strata (such as year groups) and take a random sample from each in proportion to its size: $\text{number from stratum}=\frac{\text{stratum size}}{\text{population size}}\times\text{sample size}.$ For example, 360 of 900 students in a sample of 60 gives $\frac{360}{900}\times60=24$. It gives a representative sample when the population has clear groups, but needs information about the groups.

In quota sampling the interviewer is given a quota for each group (for example 20 men and 20 women) and fills it by choosing people, often by convenience. In opportunity (convenience) sampling you take whoever is available and willing.

• Advantages: quick, cheap, no sampling frame needed; quota sampling gives a spread across groups.
• Disadvantages: not random, so likely to be biased; the researcher's choice and the time and place affect who is included; no way to measure the likely error.

To choose a method, consider whether there is a sampling frame, whether the population has distinct groups, the cost and time, and the accuracy needed. Then write a comment that refers to the context: say who might be missed or over-represented and why. Different samples from the same population give different results, so conclusions drawn from a sample are informal inferences that carry uncertainty. A larger sample generally varies less.