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(internet, newspapers, databases)	Quick to obtain; large samples available	May not be exactly what you need; may be out of date or biased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.

Example: 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.

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.

Example: Population = 500, sample size = 25. k = 500 ÷ 25 = 20. Start at a random number between 1 and 20, say 7. Select the 7th, 27th, 47th, 67th, … member.

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:

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

Number from Year 7 = (180 ÷ 900) × 50 = 10 Number from Year 8 = (200 ÷ 900) × 50 = 11.1… ≈ 11 Number from Year 9 = (190 ÷ 900) × 50 = 10.6… ≈ 11 Number from Year 10 = (160 ÷ 900) × 50 = 8.9… ≈ 9 Number from 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.

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

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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Worked Exam-Style Example

Question: 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.

Answer:

1. She is only sampling one class — students in different year groups or sets may spend different amounts of time on homework.
2. She is asking only maths students — students who take different subjects may have different homework loads. This is not a random sample of the whole school.

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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.