SamplingAQA A-Level Maths: Revision notes

The population is the whole group of people or items you want to learn about. A sample is the part of it that you actually observe. A census observes every member. Samples are used because a census may be too costly or slow, or may destroy the items tested (such as testing how long bulbs last). The list or register from which a sample is drawn is called the sampling frame. Using the sample to say something about the population is an informal inference: the sample statistic is an estimate, not the true population value.

In a simple random sample of size $n$, every member of the population has the same chance of being selected, and every possible group of $n$ members is equally likely. Method: number every member of the sampling frame, generate random numbers (calculator, computer or random number table), and pick the matching members. Reject any number that is repeated or outside the range, and carry on until $n$ different members have been chosen. Advantages: it is free from bias in the selection and every member has the same chance. Disadvantages: it needs a complete list of the population, it may be slow and costly for a large population, and a random sample can still be unrepresentative by chance.

In opportunity sampling (also called convenience sampling) you take the members of the population who are available when you collect the data, for example the first 30 people to walk past you. Advantages: quick, cheap and needs no sampling frame. Disadvantages: not every member can be chosen, so the sample is often biased. Those present at that time and place may differ from the rest of the population (for example, a shopping centre on a Saturday morning misses people working then).

Two samples from the same population usually differ. This sampling variation is why a single sample gives only an estimate. Example: of 24 000 voters, one simple random sample of 200 gives 124 supporters, a proportion of $0.62$, so about $0.62\times24000=14880$ supporters. A second sample gives 110, a proportion of $0.55$, so about $13200$. Neither is wrong; they are different estimates of the same unknown number. Larger samples are generally more reliable. A sample can also lead to a different conclusion from another sample: one may suggest a majority in favour and another may not, so be cautious about strong claims from one sample.

To choose or criticise a technique in context, ask:

• Is there a sampling frame, and can every member be chosen?
• Could the method favour some members (bias)? Say who is missing and why that matters for the question.
• Is the sample large enough to be useful?
• Is it practical in cost and time? Example: to estimate how students travel to college, asking the first 30 arrivals at the entrance is opportunity sampling and may over-represent those who live nearby. A simple random sample from the college register avoids this, but needs the register and more time. Always answer in context: name the group that is left out and say how that could change the result.