Populations, Samples and Sampling Revision Notes | GCSE AQA Higher Maths

1. Why Populations and Samples Matter

Imagine a school wants to know the average time its 1,200 students spend on homework each night. Asking every single student would take days. There has to be a smarter way — and that is exactly where the idea of a sample comes in.

Statistics is built on the idea that you can learn about a large group by studying a carefully chosen smaller group. Getting this right — knowing what counts as the full group and how to pick a fair smaller group — is the foundation of all data handling at GCSE and beyond.

2. Populations: The Full Picture

The population is defined by the question being asked. If a researcher wants to know about reading habits of Year 11 students in England, the population is every Year 11 student in England — potentially hundreds of thousands of people. The population is always the complete group relevant to the investigation.

Collecting data from an entire population is called a census. The UK government runs a national census every ten years, asking every household in the country the same set of questions. This gives perfectly accurate data — but it is enormously expensive and time-consuming. For most everyday investigations, a census is not realistic.

3. Samples: A Practical Alternative

Because studying an entire population is often impractical, statisticians select a sample — a smaller group drawn from the population. The goal is for the sample to be representative, meaning it reflects the population's characteristics as closely as possible.

Returning to the light bulb example: testing every bulb would destroy the entire day's production — clearly not an option. Instead, the factory might test 200 bulbs chosen from across the production line. If those 200 bulbs last an average of 1,000 hours, the factory can reasonably conclude that the full batch performs similarly.

4. Why Use a Sample? Advantages and Limitations

There are clear practical reasons to use a sample rather than a census. Studying a sample is faster, cheaper, and sometimes the only option — you cannot test every parachute before it is used, for example. Samples also allow investigations to happen when the population is very large or spread across a wide area.

• Faster to collect: fewer people or items to contact
• Cheaper: less time, fewer resources needed
• Necessary when testing destroys the item (e.g. crash-testing cars, testing food for contamination)
• Practical when the population is very large or geographically spread out

However, samples come with a key risk: if the sample is not chosen carefully, it may not represent the population accurately. This is called bias. A biased sample leads to conclusions that do not reflect the true picture of the population.

For example, if a school surveys only students in the top maths set to find out how much homework all students do, the results will likely overestimate the amount of homework done across the whole school. The sample does not reflect the full range of students.

Worked example: Identifying Population, Sample, and Potential Bias

A supermarket wants to know how satisfied its customers are. It surveys 50 customers who shop on a Tuesday morning. Identify the population, the sample, and one potential source of bias.

1. Population = all customers who shop at the supermarket — The investigation is about customer satisfaction across all shoppers, not just those on one day. [M1]
2. Sample = the 50 customers surveyed on Tuesday morning — This is the smaller group actually studied. [M1]
3. Tuesday morning shoppers may not represent all customers — they are more likely to be retired or not working full-time — People who work Monday to Friday are unlikely to be included, so the sample under-represents working-age customers. This makes the sample biased. [A1]

Answer: Population: all supermarket customers. Sample: 50 Tuesday morning shoppers. Bias: the sample excludes customers who work weekday mornings, so it does not represent the full population.

5. What Makes a Good Sample?

A good sample needs to be representative — it should reflect the variety and proportions found in the full population. Two factors matter most: the size of the sample and the method used to select it.

Larger samples tend to be more reliable because they reduce the effect of random variation. A sample of 10 people from a school of 1,000 could easily be unrepresentative by chance. A sample of 200 is far less likely to be skewed by a few unusual responses.

The selection method also matters. If every member of the population has an equal chance of being chosen, the sample is less likely to be biased. Selecting only volunteers, or only people who are straightforward to reach, introduces bias because certain groups are more likely to be included than others.

6. Exam Technique and Common Mistakes

GCSE Higher exam questions on populations and samples often ask you to identify the population or sample in a given scenario, explain why a sample is used, or describe a limitation of a particular sampling method. These questions reward precise language and specific reasoning tied to the context.

When asked to explain a limitation, always link your answer back to the specific scenario. Saying "the sample might be biased" on its own will not earn full marks. You need to say who is over- or under-represented and why that matters for this particular investigation.