Sampling & Bias

Sampling & Bias

When collecting data, it is usually impractical to survey every member of a population (the entire group you are interested in). Instead, you take a sample — a smaller group chosen to represent the population.

Why Sample?

• The population may be too large to survey entirely
• Saves time and money
• May be destructive (e.g. testing light bulbs until they fail — you cannot test every one)

Good and Bad Samples

A representative sample accurately reflects the characteristics of the population. A biased sample does not — it over-represents or under-represents certain groups.

Sources of bias:

• Surveying at a particular time or place (e.g. outside a gym — overrepresents fit people)
• Asking a leading question (e.g. "Don't you think...?")
• Self-selection (only people who feel strongly respond)
• Too small a sample
• Non-random selection (choosing friends)

Types of Sampling

Random Sampling

Every member of the population has an equal chance of being selected. Methods include drawing names from a hat, using a random number generator, or assigning numbers and selecting randomly.

Advantage: Eliminates selection bias

Disadvantage: Need a complete list of the population; may still get an unrepresentative sample by chance

Systematic Sampling

Select every kth member from an ordered list (e.g. every 10th person).

Advantage: Simple to carry out; evenly spread

Disadvantage: Can introduce bias if the list has a pattern that matches the sampling interval

Stratified Sampling

The population is divided into strata (groups sharing a characteristic — e.g. year group, gender). You then take a proportional sample from each stratum.

Number from each stratum = (stratum size ÷ population size) × sample size

Worked Example: A school has 200 students in Year 10 and 300 in Year 11. A stratified sample of 50 is needed.

• Year 10: (200/500) × 50 = 20 students
• Year 11: (300/500) × 50 = 30 students

Then select the required number randomly from each year group.

Advantage: Guarantees representation from each group

Disadvantage: Need to know the size of each stratum; more complex

Convenience (Opportunity) Sampling

Select whoever is easiest to reach (e.g. first 20 people you see).

Advantage: Quick and easy

Disadvantage: Very likely to be biased

Questionnaire Design

When writing survey questions, avoid:

• Leading questions: "Do you agree that..." pushes a particular answer
• Overlapping categories: response boxes like 1–5, 5–10 — where does 5 go?
• Missing categories: no option for some possible answers
• Vague time frames: "Do you exercise regularly?" — define "regularly"
• Personal or embarrassing questions without good reason

Good practice:

• Use clear, specific time frames
• Include non-overlapping, exhaustive response boxes
• Offer a "prefer not to say" option where appropriate
• Pilot the questionnaire on a small group first

Worked Example of a bad question and its improvement:

Bad: "How many hours do you spend on homework? □ 0–2 □ 2–4 □ 4+" (overlapping at 2 and 4)

Better: "In a typical week, how many hours do you spend on homework? □ 0–1 □ 2–3 □ 4–5 □ 6 or more"

Exam Tips

• When asked to criticise a sampling method, state the bias AND explain why (e.g. "people outside a gym are more likely to exercise, so the sample over-represents active people")
• Stratified sampling questions nearly always require a calculation — show the full working
• For questionnaire questions, suggest a specific improvement, not just "it is biased"
• The word "random" in maths means every member has an equal chance — it does not mean haphazard or without thought