Types of Data and Sampling

Understanding data types and sampling methods is the foundation of all statistics work in GCSE Mathematics. Before you can analyse data, you need to know what kind of data you are dealing with and how to collect it reliably.

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Types of Data

Qualitative vs Quantitative

Type	Definition	Examples
Qualitative (categorical)	Non-numerical; describes qualities or categories	Favourite colour, type of car, eye colour
Quantitative	Numerical; can be measured or counted	Height, number of siblings, temperature

Discrete vs Continuous

Quantitative data is further divided:

Type	Definition	Examples
Discrete	Can only take specific values (often whole numbers); you count it	Number of pets (0, 1, 2, 3, …), shoe size (5, 5.5, 6, …), dice score
Continuous	Can take any value within a range; you measure it	Height (172.4 cm), time (13.562 s), mass (68.3 kg)

Key tip: Ask yourself "Can the value be 3.7 and anything in between?" If yes, it is continuous.

Primary vs Secondary Data

Type	Definition	Advantages	Disadvantages
Primary	Data you collect yourself (surveys, experiments, observations)	Tailored to your purpose; you control accuracy	Time-consuming and expensive
Secondary	Data collected by someone else (internet, newspapers, databases)	Quick to obtain; large samples available	May not be exactly what you need; may be out of date or biased

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The Population and Samples

• Population: The entire group you want to find out about (e.g. all Year 11 students in England).
• Census: Collecting data from every member of the population.
• Sample: A smaller group selected from the population.

Why Sample?

• A census is often impractical: too expensive, too time-consuming, or impossible (e.g. testing every light bulb would destroy them all).
• A good sample is representative of the population — it should reflect the characteristics of the whole group.

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Sampling Methods

1. Simple Random Sampling

Every member of the population has an equal chance of being selected.

How to do it:

1. Assign a number to every member of the population.
2. Use a random number generator (or lottery method) to select your sample.

Worked Example 1

A school has 1,200 students. The head teacher wants a sample of 60. She assigns each student a number from 0001 to 1200 and uses a random number generator to pick 60 numbers.

Solution: Each of the 1,200 students has probability 60 ÷ 1200 = 1/20 of being selected. Because every student has an equal chance, the sample is unbiased. The head teacher must obtain all 60 selected students' responses — if some refuse, the sample becomes biased through non-response.

Advantages	Disadvantages
Free from bias	Need a complete list of the population (sampling frame)
Easy to understand	May not be representative if sample is small

2. Systematic Sampling

Select every kth member from a list after a random start.

How to do it:

1. Calculate k = population size ÷ sample size.
2. Choose a random starting point between 1 and k.
3. Select every kth member from that point.

Worked Example 2

Population = 500, sample size = 25. Choose a systematic sample.

Solution:

• k = 500 ÷ 25 = 20
• Choose a random starting number between 1 and 20. Suppose the random number is 7.
• Select members 7, 27, 47, 67, …, up to 487.
• Check: from 7, there are 25 terms with common difference 20. The 25th term is 7 + 24 × 20 = 487. ✓

Advantages	Disadvantages
Simple and quick	Need a list; if there is a pattern in the list, results may be biased

3. Stratified Sampling

The population is divided into strata (groups) based on a characteristic (e.g. year group, gender). A proportional sample is taken from each stratum.

Formula:

Number from stratum = (number in stratum ÷ total population) × sample size

Worked Example 3

A school surveys 50 students. The year groups are:

Year group	Number of students
Year 7	180
Year 8	200
Year 9	190
Year 10	160
Year 11	170
Total	900

Solution:

• Year 7: (180 ÷ 900) × 50 = 10
• Year 8: (200 ÷ 900) × 50 = 11.1… ≈ 11
• Year 9: (190 ÷ 900) × 50 = 10.6… ≈ 11
• Year 10: (160 ÷ 900) × 50 = 8.9… ≈ 9
• Year 11: (170 ÷ 900) × 50 = 9.4… ≈ 9

Check: 10 + 11 + 11 + 9 + 9 = 50 ✓

Within each stratum, students are selected using a random method.

Worked Example 4

A factory employs 80 line workers, 15 supervisors and 5 managers. A stratified sample of 20 is required.

Solution:

• Total population = 80 + 15 + 5 = 100
• Line workers: (80 ÷ 100) × 20 = 16
• Supervisors: (15 ÷ 100) × 20 = 3
• Managers: (5 ÷ 100) × 20 = 1
• Check: 16 + 3 + 1 = 20 ✓

Worked Example 5

A survey of 400 people gave a stratified sample of 60. One stratum (18–25 year olds) contained 120 people in the original population. How many 18–25 year olds are in the sample?

Solution: Number in sample = (120 ÷ 400) × 60 = 0.3 × 60 = 18 people.

Advantages	Disadvantages
Guarantees proportional representation	Need to know the strata sizes
More representative than simple random	More complex to organise

4. Other Methods (Brief Awareness)

• Opportunity (convenience) sampling: Choose whoever is available (e.g. asking people in a shopping centre). Quick but often biased.
• Quota sampling: Like stratified but non-random — the researcher fills quotas for each group.

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Bias

Bias means the sample is not representative of the population — certain groups are over- or under-represented.

Common Sources of Bias

Source	Example
Non-random selection	Only surveying your friends
Time of collection	Surveying a high street at 10 a.m. on Monday misses workers
Leading questions	"Don't you agree that …?" pushes people towards an answer
Non-response	People who don't reply may have different views
Too small a sample	Fewer people = less likely to be representative
Location bias	Only surveying at one location

Worked Example 6

Priya wants to find out how much time students at her school spend on homework. She decides to ask all 30 students in her maths class. Give two reasons why this may not give a representative sample, and suggest an improvement.

Solution:

1. She is only sampling one class — students in different year groups may spend different amounts of time on homework. The sample is not representative of the whole school.
2. Her class is likely all the same year group and ability set, so views may be similar. This is not a random sample of the school.

Improvement: Use stratified random sampling across all year groups to get a proportional sample of ~30 students, selected randomly within each year.

Worked Example 7

A researcher surveys shoppers outside a supermarket at 11 a.m. on a Tuesday to estimate the average household's weekly shop. State two sources of bias.

Solution:

1. Time bias: 11 a.m. on a Tuesday excludes people who work typical office hours — the sample will over-represent retirees, shift workers and people at home with children.
2. Location bias: Surveying at one supermarket misses customers of other supermarkets or those who shop online. The sample therefore does not represent all households.

Edexcel exam tip: A common question asks you to criticise a sampling method. Look for: too small, not random, only one location, leading questions, or a specific time that excludes groups.

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Common Mistakes

• Confusing discrete and continuous — shoe sizes (3, 3.5, 4, …) are discrete even though they include halves; they are fixed values you count through.
• Forgetting to round stratified sample sizes correctly. If the total does not match the required sample size, adjust the stratum closest to 0.5.
• Saying "the sample is biased because it is too small" without explaining why a small sample might not represent the population.

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Practice

1. Classify each as qualitative or quantitative: (a) colour of eyes; (b) temperature; (c) number of goals scored.
2. State whether each is discrete or continuous: (a) number of text messages; (b) time taken to run 100 m; (c) GCSE grade (9–1).
3. A factory has 80 line workers, 15 supervisors and 5 managers. A stratified sample of 20 is needed. How many of each group should be selected?
4. Explain one advantage and one disadvantage of using a questionnaire delivered online.

Answers:

1. (a) Qualitative; (b) Quantitative; (c) Quantitative.
2. (a) Discrete; (b) Continuous; (c) Discrete.
3. Line workers: (80/100) × 20 = 16; Supervisors: (15/100) × 20 = 3; Managers: (5/100) × 20 = 1. Total = 20 ✓.
4. Advantage: Can reach a large number of people quickly and cheaply. Disadvantage: Non-response bias — only people with internet access and willingness to respond will complete it, so it may not represent the whole population.

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Summary

• Data can be qualitative or quantitative (discrete or continuous).
• Primary data is collected first-hand; secondary data comes from existing sources.
• A sample should be representative. Key methods: random, systematic, stratified.
• Stratified sampling uses the formula: (stratum size ÷ population) × sample size.
• Bias occurs when a sample systematically favours some groups. Look out for non-random selection, leading questions, small samples and poor timing.

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Extended Worked Examples

Worked Example A — Stratified sampling with three strata

A secondary school has students across three year groups: Year 9 has 240 students, Year 10 has 180 students and Year 11 has 180 students. The head of year wants a stratified sample of 50 students to take part in a lesson-time survey. How many students should be chosen from each year group?

Step 1: Find the total population: $240 + 180 + 180 = 600$240+180+180=600.

Step 2: Find the sampling fraction: $\frac{50}{600} = \frac{1}{12}$60050​=121​.

Step 3: Apply the fraction to each stratum.

Year group	Population	Calculation	Sample
Year 9	240	$240 \times \frac{1}{12}$240×121​	20
Year 10	180	$180 \times \frac{1}{12}$180×121​	15
Year 11	180	$180 \times \frac{1}{12}$180×121​	15
Total	600		50

Check: $20 + 15 + 15 = 50$20+15+15=50. Always add the strata back together as a final sanity check — many students lose a mark through arithmetic slips at this stage.

Callout — rounding rule: If a calculation gives a non-integer (e.g. 14.6), round to the nearest whole number, but check the total still equals the required sample size. If not, adjust by one in the strata where rounding was closest.

Worked Example B — Sampling frame and bias

A town council wants the opinions of residents about a new cycle lane. They stand at a bus stop at 10am on a Tuesday and ask the first 60 people who pass.

Discuss whether this produces a representative sample.

Solution: The sample is biased for several reasons:

1. Time of day: People in work at 10am (most of the working population) cannot be surveyed.
2. Location: Only people near a bus stop are picked — cyclists and drivers are systematically excluded, even though they are key stakeholders for a cycle lane.
3. Self-selection: People who pass that bus stop may be a particular demographic (e.g. retired people, shift workers).

A better method would be a stratified random sample taken from the full electoral roll, using categories such as age band and mode of transport so that each group is proportionally represented.

Worked Example C — Capture-recapture (Higher tier)

A biologist catches 80 fish from a lake, tags them, and releases them. One week later she catches 100 fish, of which 16 are tagged. Estimate the total population of fish in the lake.

Solution: Using the capture-recapture formula $\frac{\text{tagged in sample}}{\text{total sample}} = \frac{\text{total tagged}}{\text{population}}$total sampletagged in sample​=populationtotal tagged​:

$\frac{16}{100} = \frac{80}{N}$10016​=N80​

Rearranging: $N = \frac{80 \times 100}{16} = \frac{8000}{16} = 500$N=1680×100​=168000​=500 fish.

Assumptions: No fish entered or left the lake; tags did not fall off; tagged fish mix uniformly with untagged fish; tagged and untagged fish are equally likely to be caught.

Worked Example D — Bias in survey questions

Rewrite the question: "Don't you agree that the new uniform is much better than the old one?" to remove bias.

Solution: The question is leading (it suggests the "correct" answer). A neutral version:

"How do you rate the new uniform compared with the old one? Much better / Slightly better / About the same / Slightly worse / Much worse."

Balanced response options with equal positive and negative choices reduce the risk of skewing results.

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Answering at different grade levels

Exam-style question (4 marks): A gym has 2,400 members. The owner wants a stratified sample of 60 members across age bands. There are 600 under-25s, 1,200 members aged 25–50 and 600 over-50s. Work out how many members from each band should be chosen and explain why stratified sampling is appropriate.

• Grades 3–4 answer: "60 ÷ 2400 = 0.025. Under-25: 600 × 0.025 = 15. 25–50: 1200 × 0.025 = 30. Over-50: 600 × 0.025 = 15." (Correct arithmetic but no explanation — gains method and accuracy marks only.)
• Grades 5–6 answer: "Sampling fraction = 60/2400 = 1/40. Under-25: 600 × 1/40 = 15. 25–50: 1200 × 1/40 = 30. Over-50: 600 × 1/40 = 15. Check: 15 + 30 + 15 = 60. Stratified sampling keeps the correct proportion of each age band." (Correct working and a basic reason.)
• Grades 7–9 answer: "Using stratified random sampling with fraction 60/2400 = 1/40 gives: under-25 → 15, 25–50 → 30, over-50 → 15 (total 60). Stratified sampling is appropriate because membership is not evenly distributed across age bands; a simple random sample could under-represent the smaller strata by chance. Within each stratum members must be chosen by simple random sampling to avoid selection bias." (Full marks: correct figures, check, and precise justification using the terms bias, stratum and simple random sampling.)

Edexcel alignment: This content is aligned with Edexcel GCSE Mathematics (1MA1) specification — specifically Topic S (S1 Sampling). Assessed on Papers 2 and 3.