Methods of samplingEdexcel International A Level Further Maths: Revision notes

A population is the whole set of items or individuals under investigation. A census observes or measures every member of the population. A sample observes only some of them. The list of all the sampling units in the population is the sampling frame, and each member of that list is a sampling unit. A census gives the true value of any population parameter and has no sampling error, but it is expensive, slow, and impossible if testing destroys the item (for example testing how long light bulbs last) or if the population is infinite. A sample is cheaper, quicker and the only choice for destructive testing, but its results are subject to sampling error, so conclusions are less reliable and there is a risk of bias if the sample is not representative. A good sampling frame is complete, up to date and free of duplicates. Missing or duplicated units in the frame make the sample biased before any selection has been made.

In a simple random sample of size $n$, every possible sample of $n$ units is equally likely to be chosen (so every unit has the same chance of selection). The method needs a sampling frame. To take one using random numbers:

• number the units in the sampling frame, using the same number of digits for each (for example 001 to 500);
• generate random numbers (calculator, computer or table) with that many digits;
• ignore any number that does not match a unit, and ignore repeats, because the sample is taken without replacement;
• continue until $n$ different units have been chosen. Advantages: free of selection bias, and sampling error can be estimated. Disadvantages: it needs a complete sampling frame, which may be costly to produce, and with a large population it can be time-consuming. A small random sample may also, by chance, not represent the population well, for example missing a small group altogether.

A systematic sample takes every $k$th unit from the sampling frame. The sampling interval is $k=\frac{N}{n}$, where $N$ is the population size and $n$ the sample size. The first unit is chosen at random from the first $k$ units, then every $k$th unit after it is selected. For a list of 840 students and a sample of 60, $k=\frac{840}{60}=14$. If the random start is 9, the sample is the 9th, 23rd, 37th, ... names. Advantages: simple and quick to carry out, and the sample is spread evenly through the list. It needs only a random start, not a random number for every unit. Disadvantages: it needs a sampling frame, and it is not a simple random sample, because not every sample is possible (for example two neighbouring units can never both be chosen). If the list has a regular pattern whose period matches $k$, the sample can be badly biased.

When the population divides into distinct groups (strata) such as year groups or departments, a stratified sample takes a simple random sample from each stratum, in proportion to its size: $\text{number from a stratum}=\frac{\text{stratum size}}{\text{population size}}\times n.$ For 480 employees (192, 168 and 120 in three departments) and $n=40$, the sampling fraction is $\frac1{12}$, giving 16, 14 and 10. Advantages: every stratum is represented in the correct proportion, so the sample is more representative than a simple random sample, and it is especially useful when the strata differ in the characteristic being studied. Disadvantages: the population must be classified into strata with a complete list of each, which is more time-consuming and costly, and rounding the numbers from each stratum can make the proportions slightly inexact.

In a quota sample the population is divided into groups (such as age and gender) and an interviewer chooses people until a fixed quota in each group is filled, with quotas proportional to the group sizes in the population. It is non-random: the interviewer decides whom to approach. Advantages: no sampling frame is needed, it is quick and cheap, a refusal is simply replaced so there is no non-response problem, and each group is guaranteed to be represented. Disadvantages: because it is not random, there can be interviewer bias and selection bias (for example only interviewing people who look friendly or who are in a particular place at a particular time), and the sampling error cannot be estimated. It is often used in market research and street surveys where a sampling frame is not available.

Match the method to the context:

• Complete list available and a small, uniform population: simple random (random numbers).
• Long ordered list and speed matters: systematic, checking for a pattern in the list.
• Distinct groups that may differ in what is measured: stratified.
• No list available (street or shop survey) and a quick, cheap answer needed: quota.
• Destructive testing or an infinite population: a sample, never a census. In an evaluation question give a point for the method and a point for the context, state a limitation, and then reach a justified conclusion. A strong limitation is specific to the scenario, for example 'only shoppers on a Saturday morning could be interviewed'.