Populations and Sampling
How do companies know if you’ll like a new snack before it hits the shelves? How do political parties predict election results without asking everyone? The answer is sampling — making smart inferences about a big group by studying a carefully chosen small part of it.
Population vs. Sample
The Population
The population is the entire group you are interested in. In a survey of your school’s favourite crisps, the population is all the students in the school.
The Sample
A sample is the smaller group you actually collect data from. Instead of asking every student, you might pick 50 — those 50 are your sample.
We use samples because collecting data from an entire population is often impossible, too expensive, or too slow. We use the sample to make an inference about the whole population.
The Two Big Limitations: Bias and Sample Size
Bias
Bias means your sample does not accurately represent the population. Common causes:
- Convenience sampling — picking people who are easy to reach (e.g., just your friends).
- Excluding groups — only surveying people in one location or age range.
- Leading questions — phrasing that nudges people towards a certain answer.
- Self-selection — only people with strong opinions choose to respond.
Sample Size
The sample size is the number of individuals in your sample.
- A very small sample may not reflect the population, even if randomly selected.
- A larger sample generally gives more reliable results — as long as it is also unbiased.
- A large but biased sample still produces wrong conclusions — just with more confidence.
Key point: For GCSE, the emphasis is that a very small sample often leads to unreliable conclusions, and that both size and fairness of selection matter.
Key Terms
Population
The entire group of items or individuals being studied.
Sample
A smaller, selected group taken from the population, used to make inferences about the whole.
Inference
A conclusion or generalisation about a population based on data from a sample.
Bias
When a sample does not accurately represent the population, leading to misleading conclusions.
Random Sample
A sample where every member of the population has an equal chance of being selected.
Sample Size
The number of individuals or items in a sample. Larger is generally more reliable, if unbiased.
Worked Examples
Example 1: Identifying Population and Sample
A local council wants to find out what residents think about a new recycling centre. They survey 200 randomly selected homes.
- Population: All residents (or all homes) in the council’s area — the full group they want to learn about.
- Sample: The 200 randomly selected homes that received the survey — the group data was actually collected from.
Example 2: Spotting Bias
A newspaper asks “Do you agree that the King is doing a fantastic job?” via a poll on their website.
- Self-selection bias: Only visitors to that newspaper’s website who feel strongly enough will respond — not representative of the general public.
- Leading question: The phrasing suggests a positive answer, pressuring agreement and failing to capture neutral or negative views.
Effect: The results will reflect only a specific type of reader, not the wider public’s opinion.
Example 3: Sample Size Limitations
A pizza restaurant asks the first 5 customers about a new pizza. All 5 say it’s “amazing!” and the manager declares it a future best-seller.
- Limitation: A sample of just 5 customers is far too small. These first few customers may love spicy food, know the manager personally, or simply be polite — they are unlikely to represent the full customer base.
- Improvement:
- Survey a much larger group (e.g., 50–100 customers over a week).
- Use a neutral rating scale (e.g., 1–5 stars) rather than asking a leading question.
- Distribute feedback forms with every order to capture a wider range of customers systematically.
Tutor Insights
🤔 Common Misunderstandings
- Mixing up population and sample. Population = everyone you’re interested in. Sample = who you actually ask.
- Thinking “random” means “any old way.” True random sampling means every member of the population has an equal chance of selection.
- Bigger is always better. A large but biased sample still gives wrong results — just confidently wrong ones. Quality of selection matters as much as quantity.
📝 Common Exam Mistakes
- Vague explanations: Don’t say “they should ask more people” — say “they should ask a larger, random sample from different age groups and locations.”
- Not giving context for bias: Just writing “it’s biased” won’t score marks. Explain why it leads to a non-representative sample.
- Forgetting how to select randomly: If asked to improve a method, suggest a practical technique — e.g., “use a random number generator to pick student ID numbers.”
Practice Questions
- A wildlife group wants to estimate the average weight of foxes in a large forest. They catch and weigh 30 foxes.
(a) What is the population? (b) What is the sample? - A shop owner asks customers to write “Yes” or “No” on a piece of paper at the till about whether they’d like longer opening hours.
(a) Suggest one reason why this might be biased. (b) How could this affect her conclusion? - A teacher asks the first 5 students to arrive one morning about their homework habits.
(a) Why might this sample not be representative of all Year 9 students?
(b) Suggest two ways to get a more reliable sample. - A company selects 10 bulbs from a batch of 5,000 and tests them to failure.
(a) Why is it not practical to test all 5,000?
(b) Is a sample of 10 likely to be a good representation? Explain. - A student asks their 10 friends what new extracurricular activities a school of 800 students should introduce.
(a) Give two reasons why this is a biased method.
(b) Suggest a better sampling approach, justifying your choice.
Show Answers
- (a) All foxes in the large forest. (b) The 30 foxes caught and weighed.
- (a) Self-selection bias — only customers who notice the sign and feel strongly enough will respond. Customers in a hurry or those without strong opinions won’t participate.
(b) She may overestimate demand for longer hours, as only those who really want them bother to respond. - (a) Convenience sample with a tiny sample size. The first five to arrive may be punctual, motivated students — not representative of all Year 9 habits. Five is far too small a sample.
(b) 1) Increase the sample size significantly (e.g., 50 students). 2) Select students randomly from the full Year 9 register (e.g., using a random number generator). - (a) Destructive testing — testing a bulb until it fails means you can no longer sell it. Testing all 5,000 would destroy the entire stock.
(b) No — 10 is too small a sample relative to 5,000. By chance, those 10 could be unusually long- or short-lasting. A larger random sample would be more reliable. - (a) 1) Friendship/convenience bias — 10 friends likely share similar interests, not reflecting 800 diverse students. 2) Sample size too small — 10 out of 800 (1.25%) is not representative.
(b) Obtain a list of all 800 student IDs and use a random number generator to select 50–80 students. This gives every student an equal chance of inclusion, producing a representative sample.
FAQs
Q: What’s the difference between population and sample?
A: The population is the entire group you want to study (e.g., all teenagers in the UK). The sample is the smaller group you actually collect data from (e.g., 100 randomly selected teenagers).
Q: Why is a random sample important?
A: It ensures every member of the population has an equal chance of being selected, making the sample more likely to represent the whole population and reducing bias.
Q: Does a bigger sample always mean better results?
A: Generally yes — if the sample is unbiased. A large but biased sample just gives you a confidently wrong answer. Quality of selection is as important as quantity.
Q: How can I tell if a sample is biased?
A: Ask: was every member of the population equally likely to be chosen? Was the method convenient but not fair? Were the questions neutral? If not, the sample is likely biased.
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