Making Inferences from Samples Revision Notes | GCSE AQA Higher Maths

1. Why Samples Matter

Imagine a school wants to know the average time its 1,200 students spend on homework each night. Asking every single student is slow, expensive, and often impractical. Instead, the school surveys 60 students and uses those results to draw conclusions about everyone. That process — using a sample to say something about a population — is called making an inference.

This is one of the most powerful ideas in statistics, and it appears directly on the Higher-tier GCSE paper. The key skill is not just reading off a sample statistic; it is knowing when you can trust it and when you cannot.

2. Populations and Samples

A sample is a smaller group chosen from the population. The sample is what you actually measure. The goal is for the sample to be representative — meaning its characteristics reflect those of the whole population as closely as possible.

The link between sample and population is the heart of this topic. A sample statistic — such as a sample mean or a sample proportion — is used as an estimate of the corresponding population value. The reliability of that estimate depends entirely on how the sample was chosen.

3. Making Inferences: The Core Method

When you have sample data, you can use it to estimate population values. The most common inferences at GCSE involve estimating a population proportion (a fraction or percentage) or a population mean. The method is to treat the sample statistic as your best estimate for the population.

Worked example: Making an Inference from a Sample Proportion

A factory produces bolts. A quality inspector takes a random sample of 200 bolts and finds that 14 are defective. Estimate the number of defective bolts in a batch of 5,000.

1. $\text{Proportion defective} = \frac{14}{200} = 0.07$ — Divide the number of defectives by the total sample size. This gives the fraction defective in the sample. [M1]
2. $\text{Estimated defectives in 5,000} = 0.07 \times 5000$ — We assume the sample proportion is a reliable estimate of the population proportion, so we scale it up. [M1]
3. $0.07 \times 5000 = 350$ — Multiply: $0.07 \times 5000 = 350$. [A1]
4. Estimated number of defective bolts in the batch: $350$ — Always write the answer in the context of the question. Missing context loses the communication mark.

Answer: An estimated 350 bolts in the batch of 5,000 are defective.

Notice the word estimated in the final answer. That word matters. You are not claiming the population value is exactly 350 — you are inferring it based on sample evidence. Examiners reward students who use language that acknowledges this uncertainty.

Worked example: Making an Inference from a Sample Mean

A researcher measures the heights of a random sample of 10 plants from a greenhouse containing 400 plants. The heights (in cm) are: 12, 15, 14, 16, 13, 17, 14, 15, 13, 11. Estimate the total height of all 400 plants.

1. $\text{Sum} = 12 + 15 + 14 + 16 + 13 + 17 + 14 + 15 + 13 + 11 = 140$ $\text{Sample mean} = \frac{140}{10} = 14 \text{ cm}$ — Add all values: $12+15=27$, $27+14=41$, $41+16=57$, $57+13=70$, $70+17=87$, $87+14=101$, $101+15=116$, $116+13=129$, $129+11=140$. Divide by 10. [M1]
2. $\text{Estimated population mean} = 14 \text{ cm}$ — We use the sample mean as our best estimate of the mean height of all 400 plants. [M1]
3. $\text{Estimated total} = 14 \times 400 = 5600 \text{ cm}$ — Multiply the estimated mean by the population size: $14 \times 400 = 5600$. [A1]
4. Estimated total height of all 400 plants: $5600$ cm — Include units and context in your final answer.

Answer: The estimated total height of all 400 plants is 5,600 cm.

4. Limitations of Sampling

Here is where the topic gets more demanding — and where the exam marks are often dropped. A sample can only produce a reliable inference if it is representative of the population. Several factors can prevent this.

• Sample size too small: A small sample is more likely to produce results that differ from the population by chance. Larger samples generally give more reliable estimates.
• Sampling bias: If the method of selecting the sample favours certain individuals over others, the sample will not reflect the population fairly.
• Non-random selection: Convenience sampling (e.g. asking only your friends) or voluntary response (e.g. online polls) can produce highly unrepresentative samples.
• The population may change: If the population changes over time, a sample taken at one point may not reflect the population at a later date.
• Variability: Even a perfectly random sample will differ from the population to some degree, due to chance. Different samples from the same population will give different results.

These limitations mean that inferences are always estimates, not certainties. The larger and more randomly selected the sample, the more confidence you can place in the inference — but you can never be completely certain that the sample perfectly mirrors the population.

5. Worked Exam Question: Evaluating an Inference

GCSE Higher exam questions on this topic often ask you to evaluate a claim made from a sample. That means you need to state whether the inference is reliable and explain why or why not, using the context of the question. Here is a typical example.

Worked example: Evaluating the Reliability of a Sample Inference

Mohammed surveys 5 students from his year group of 180 to find out how many hours per week they spend gaming. He finds a mean of 9 hours and concludes: 'Students in my year group spend an average of 9 hours per week gaming.' Evaluate this conclusion.

1. Mohammed is using a sample mean of 9 hours to infer the population mean for all 180 students. — Name the inference clearly before evaluating it.
2. $\text{Sample size} = 5$ out of $180$ students $= \frac{5}{180} \approx 2.8\%$ of the population — A sample of only 5 from 180 is very small. Small samples are more likely to produce results that differ significantly from the population by chance. [M1]
3. We are not told how the 5 students were selected. If Mohammed chose friends or students nearby, the sample may not be representative of all 180 students. — Non-random selection introduces bias, which undermines the reliability of the inference. [M1]
4. The conclusion is unreliable because the sample is too small and the selection method is unknown, so the sample may not represent the year group. — Always link your reasoning back to the specific context of the question. [A1]

Answer: Mohammed's conclusion is unreliable. The sample of 5 is too small to represent 180 students, and without knowing the selection method, the sample may be biased. A larger, randomly selected sample would produce a more reliable inference.

That final answer earns marks because it does three things: it states a judgement ('unreliable'), gives a reason linked to sampling theory ('too small', 'may be biased'), and suggests an improvement ('larger, randomly selected sample'). Practise structuring your answers this way.

6. Exam Technique and Common Mistakes

This topic rewards students who write precisely. The difference between a 1-mark answer and a 3-mark answer is usually the depth of reasoning and the use of correct statistical language. The guidance below consolidates the most important exam-technique points for this topic.

Bringing this all together: every time you see a sample result in an exam question, ask yourself two things. First, can I use this to estimate a population value? Second, how reliable is that estimate, given what I know about the sample? Those two questions cover everything higher-tier exams can ask on this topic.