Data, Sampling and Population Estimates
Statistics uses data to answer questions about a group. A useful conclusion depends on what was measured, who was included and how the data were collected—not just on the calculation.
Foundation & Higher, with a Higher capture–recapture extension. You will use fractions, decimals and percentages and direct proportion to turn sample results into estimates.
Define the group and the question
A population is the whole group your question concerns. A sample is the part you actually study. A census collects information from every member of the population. A sample usually costs less time and money, but it introduces uncertainty.
Worked example — make the population precise
To investigate journeys to school, “everyone” is too vague. A suitable population is all 960 students enrolled at the school this term. A sample of 80 students gives information about those 960 students; it does not automatically describe every school in the country.
Decide what to record before collecting data: for example, main travel method and journey time in minutes on a normal school day. Measure each quantity consistently. A census avoids sampling variation, but incorrect measurements or missing replies can still make it inaccurate.
Identify the type and source of data
Primary data are collected first-hand for your investigation. Secondary data already exist, such as an earlier survey or published records. Check whether their date, definitions and population match your question.
| Data | Type |
|---|---|
| Travel method | Categorical: labels |
| Number of siblings | Discrete: a count |
| Journey time | Continuous: a measurement |
Numerical data are also called quantitative; categorical data are often called qualitative. A discrete variable takes separate possible values, usually whole-number counts. A continuous variable can take any value in a range, even if the recorded measurement is rounded.
Worked example — recording does not change the variable
Height rounded to the nearest centimetre is still a continuous measurement. A house number is a label, not a numerical quantity for which an average would be meaningful.
Select a simple random sample
In a simple random sample, every possible sample of the chosen size has the same chance of selection. A practical method is to number a complete list of the population and use a random-number generator.
Worked example — choose 40 from a list of 600
Label the members 1 to 600. Generate numbers from 1 to 600 and select the corresponding members. Ignore repeats until you have 40 different numbers. Do not replace inconvenient people with friends: this changes the selection method.
Picking whoever is easiest to reach is not the same as random selection. Even a genuinely random sample can differ from the population by chance. A larger well-selected sample usually reduces this sampling variation; it cannot guarantee a perfectly representative result.
Find and reduce sampling bias
A representative sample reflects the relevant features of its population. Bias is a tendency in the collection method to favour particular results.
Worked example — location and non-response matter
Asking only students arriving at the cycle racks will over-represent cyclists. Select students from the whole school register instead, then seek replies across the selected group. If only enthusiastic cyclists reply, non-response can still bias the result.
Check coverage, timing and wording. A lunchtime survey may exclude people working then. “How excellent is our service?” encourages approval. Ask neutrally, use clear non-overlapping response categories and record all relevant outcomes. A large biased sample can be less useful than a smaller well-selected one.
Use a sample to estimate a population quantity
If the sample is suitably representative, apply its proportion to the population. This is an estimate, not a guaranteed count.
Worked example — scale a sample result
Of 80 randomly selected students, 18 cycle to school. The sample proportion is 18/80 = 0.225 = 22.5%. For a school of 960, the estimate is 960 × 18/80 = 216 cyclists.
This assumes the sample reflects the whole school. Different samples may give different estimates. If the sample came only from the cycle racks, the estimate would probably be too high.
A sample mean can also estimate a population mean. For example, a representative sample mean journey time of 17 minutes suggests a population mean near 17 minutes; it does not mean every journey takes 17 minutes. The Averages and Spread lesson develops these summaries.
Higher extension — capture–recapture
Edexcel Higher includes estimating a population that is difficult to count directly. Capture M individuals, mark them safely and release them. After they mix, take a second sample of n; let r of these be marked.
The model assumes the marked proportion in the second sample represents the marked proportion in the whole population: r/n ≈ M/N. Therefore the population estimate is N ≈ M × n ÷ r. This is proportional reasoning; the two sample sizes are not simply added.
Worked example — estimate pond insects
First capture, mark and release 45 insects. Later capture 40, including 9 marked insects. The estimated marked fraction is 9/40. The 45 marked insects represent that fraction of the population, so N ≈ 45 × 40 ÷ 9 = 200 insects.
Assume the population is effectively closed between samples, marks remain visible, individuals mix thoroughly, and marking does not change the chance of recapture. If marked insects become easier to catch, r may be too large and the estimate too small. If r = 0, this formula cannot give a finite estimate; gather more suitable data.
Think Like a Mathematician
Doubling a biased sample does not remove its bias. The method determines who can be represented; the sample size mainly affects chance variation once that method is suitable.
Common Mistakes
Sample versus population: state the group the question actually concerns.
Large means unbiased: size cannot repair systematic exclusion.
Estimate means fact: explain the assumption behind scaling a sample.
Capture–recapture: only marked individuals in the second sample belong in r.
Practice
1. A school has 750 students; 60 are surveyed about lunch. State the population and sample.
Population: all 750 students at that school. Sample: the 60 surveyed.
2. Classify shoe brand, number of pets and measured mass.
Shoe brand is categorical; number of pets is discrete numerical; measured mass is continuous numerical.
3. Explain how to choose a simple random sample of 25 from 500 members.
Number a complete list 1–500; use a random-number generator; ignore repeats until 25 distinct members are selected.
4. A shop surveys only people who post enthusiastic reviews. Explain one likely limitation.
It excludes many other customers and favours strongly positive opinions. It is unlikely to represent all customers.
5. In a representative sample of 50 households, 14 have a dog. Estimate how many of 1200 households have a dog.
1200 × 14/50 = 336 households. This assumes the sample proportion represents the whole population.
6. Will a random sample twice as large always give a closer estimate?
No. It usually reduces sampling variation, but any particular result may be further from the true value.
7. Higher: 60 animals are marked. A later sample of 50 includes 12 marked animals. Estimate the population.
N ≈ 60 × 50 ÷ 12 = 250 animals, assuming a closed, well-mixed population with equal capture chances.
8. Higher: some marks rub off before the second capture. What happens to the estimate, other things equal?
The recorded marked count r is too small. Dividing M × n by a smaller r makes the estimate too high.
Frequently Asked Questions
Must I learn stratified-sampling calculations here?
They are not required by the current GCSE Mathematics specifications used for this course. Pearson explicitly excludes them. Focus on random selection, representative data and justified limitations.
Is capture–recapture Probability or Statistics?
It estimates a population from sampled data, so it is taught here. It uses proportional reasoning; it does not need a tree diagram.