K: Statistical sampling

AQA A-Level Mathematics: K Statistical Sampling

Specification reference: AQA A-level Mathematics (7357), 3.12 K: Statistical sampling, K1.

Overview

Statistical sampling is the art of using a manageable part of a group to say something sensible about the whole group. In an AQA question, the mathematics is often short, but the reasoning is not: candidates must identify the population and sample, choose or criticise a method in context, and make a conclusion that is cautious rather than overconfident. The specification explicitly includes simple random sampling and opportunity sampling, then expects you to understand that two valid samples can still lead to different conclusions.

This is the first step in the statistics story. You sample before you draw diagrams or calculate summary statistics in Topic L, before you model probability in Topic M, and before you test a claim in Topic O. A typical question gives a survey, experiment or data set and asks you to state a population, describe a sampling method, explain possible bias, or assess whether a conclusion is justified. Marks are awarded for contextual precision: say who was sampled, how they were chosen, and what that means for the conclusion.

Examiner’s lens: A sample result is evidence about a population, not the population itself. Credit is given for language such as “suggests”, “may indicate” and “is not necessarily representative”.

Revision Podcast

Listen once for the overview, then pause at the quick-fire quiz and answer aloud before continuing. The audio uses the same examiner-creditworthy phrasing as the worked examples, so it is designed for active recall rather than passive listening.

Key Concepts

1. Population, sample and sampling frame

The population is the complete group about which the investigation wants information. It is defined by the question, not by how many people are conveniently available. If a university wants to know the average weekly study time of all first-year students, the population is every first-year student at that university. A sample is the smaller group from which data are actually collected.

A sampling frame is the list or source from which the sample is selected. It must match the target population as closely as possible. A register of students who attend a lunchtime club is not a sound frame for every student in the school, because it excludes non-attenders. This matters because no random-number generator can repair a frame that has already left out part of the population.

Example: A sixth form has 360 Year 12 students. A researcher uses the official Year 12 register to choose 40 students. Population = all 360 Year 12 students; sample = the 40 chosen students; sampling frame = the official register of 360 names.

2. Simple random sampling: fair selection, not guaranteed perfection

In a simple random sample, every member of the population has an equal chance of being selected. A robust exam description is: make or use a complete sampling frame, assign each member a unique number, use a random-number generator or random-number table to choose the required distinct numbers, then collect data from those corresponding members.

Why does this work? Random selection removes deliberate choice by the investigator. It reduces selection bias, because no one is picked simply for being easiest to contact or most likely to agree. However, random does not mean perfectly representative. By chance, a small random sample can contain more keen cyclists, taller students or higher earners than the population proportion. That is sampling variation, not proof that the method was wrong.

Example: To choose 25 pupils from a numbered list of 1 to 500, generate 25 distinct integers from 1 to 500. If 17, 203 and 487 appear, select the pupils numbered 17, 203 and 487. Do not keep generating until the numbers “look balanced”; that would destroy the random design.

3. Opportunity sampling: efficient but exposed to bias

An opportunity sample is made from people who are available or easy to reach. Asking the first 30 visitors leaving a sports centre is opportunity sampling. It may be quick, cheap and practical, which can be a reasonable advantage in a real investigation. The weakness is not merely that it is “unfair”. The precise criticism is that available people may differ systematically from the whole population. Sports-centre visitors may be more active than local residents generally, so their exercise habits are unlikely to represent all residents.

Use the context in your criticism. “It may be biased” earns less than “it over-represents people who use the sports centre, so the estimated weekly exercise time may be too high”. Also notice non-response: even if people are approached, those who refuse may differ from those who agree.

4. Representativeness, bias and informal inference

A representative sample resembles the relevant features of the population closely enough for the purpose of the investigation. There is no magic rule that a sample is representative simply because it is large. A very large opportunity sample from one location can still miss whole groups. A smaller, well-designed random sample may be more useful.

Bias is a systematic tendency for a sample or measurement process to favour some outcomes or groups. Selection bias can come from the frame, the method, the location, the time of day or the people who choose to respond. A leading question can introduce response bias too, although the question wording is a separate design issue.

An informal inference is a cautious conclusion from sample evidence to the population. If 36 out of 60 randomly chosen students prefer digital notes, write: “The sample suggests that around 60% of students may prefer digital notes.” Do not write: “Exactly 60% of all students prefer digital notes.” The first statement recognises that another sample could give a different result.

5. Different samples, different conclusions

Suppose two independent random samples of 20 students give mean commuting times of 28.4 minutes and 31.1 minutes. Neither result has to be wrong. A sample statistic changes from sample to sample because the people selected change. AQA expects candidates to recognise this point when they critique an overconfident population claim.

The distinction between a parameter and a statisticsharpens this idea. A population mean, often written (\mu), is a fixed but usually unknown parameter. A sample mean, (\bar{x}), is a statistic calculated from the selected observations. You may use a sample mean as evidence about (\mu), but they are not automatically equal.

5a. Turning statistical understanding into marks

Most sampling questions reward the quality of your interpretation, not a long list of definitions. Begin with the command word. For state or identify, write the exact target group or sampling method and stop. For describe, give the method as a sequence of actions. For explain, connect cause to consequence using “because”. For evaluate, include a judgement after the evidence.

If an opportunity sample is taken outside a gym, an answer such as “the sample may be biased” is a starting point only. Improve it in three moves: identify the group, link it to the variable, then qualify the conclusion. For example: “Gym users are likely to exercise more than non-users, so this opportunity sample may overestimate the population’s mean weekly exercise time.” This earns credit for method, context and effect.

Similarly, do not praise simple random sampling as “unbiased” without qualification. A more accurate statement is: “It gives every member an equal chance of selection and reduces selection bias, provided the sampling frame is complete. However, the sample may still be unrepresentative by chance.” This distinction is exactly what separates a basic answer from an AO2/AO3 answer: you are reasoning about the limits of a model in context.

When a question supplies a mean or proportion, calculate accurately and then write one sentence of interpretation. Include the unit, name the population, use cautious language and, if appropriate, state a limitation. An examiner can award credit for a concise conclusion that does all four jobs. Do not replace explanation with a calculator screen value.

6. Choosing and critiquing methods in context

A high-mark answer weighs feasibility against validity. Simple random sampling is usually stronger when a full sampling frame exists, but it may be slow or expensive to contact selected people. Opportunity sampling is fast, but it is especially vulnerable to location and time bias. Other methods can be useful comparisons: a systematic sample selects at a fixed interval after a random start; a stratified sample samples from groups in proportion to their population sizes; a cluster sample selects whole groups. These methods are useful vocabulary, but keep your answer anchored to the method and context in the question.

Mathematical Relationships and Calculator Use

There is no new formula to memorise for K1. The key relationship is conceptual:

[\text{sample statistic } \bar{x} \text{ is used to estimate the population parameter } \mu.]

When a question gives numerical data, calculator technology can calculate a sample mean, standard deviation or other summary statistic. The arithmetic is only the start. You must attach meaning: “The sample mean is 7.4 hours; this suggests the population mean may be near 7.4 hours, provided the sample is sufficiently representative.”

Formula status:(\bar{x}) as a sample mean and (\mu) as a population mean aremust-know notation, not a separate K1 formula-sheet calculation. The important skill is interpreting a calculator output as a sample statistic rather than reporting it as an exact population fact. No unit conversion is specific to this topic.

Practical Applications

Sampling underpins opinion polls, quality control, clinical research, transport surveys and the AQA large data set. A council cannot usually interview every resident about bus use, so it samples residents. A factory cannot always test every item destructively, so it tests a sample. In each case, ask the same three questions: **Who is the population? What frame was used? Could the selection over-represent some people or items?**There is no required practical for A-level Mathematics K1. Treat a survey-design question as a miniature investigation: define the population, design or assess the sampling route, calculate summaries using technology if asked, then state an inference with a limitation.