Sampling — populations and samples

Sampling sits in OCR J560 Statistics across all three calculator papers. It is largely descriptive (no formulas required at GCSE), so marks are awarded for clear reasoning, not arithmetic.

Population vs sample

• The population is the full group you want to know about (e.g. all 1500 students at a school).
• A sample is a subset chosen from the population (e.g. 60 students surveyed).

Sampling is needed because measuring the whole population is usually impractical (cost, time, destructive testing).

Properties of a good sample

1. Random: every member of the population has an equal chance of being chosen.
2. Large enough: too small and results are unreliable; too large and it costs too much.
3. Representative: reflects the structure of the population (gender, age, year group, etc.).

Random sampling methods

• Simple random sampling: number every member; use a random number generator or random number table to pick.
• Systematic sampling: pick every kth person from an ordered list (e.g. every 25th student).
• Stratified sampling (Higher): split the population into groups (strata) and pick from each in proportion. E.g. if 60% of the school are girls, take 60% of your sample as girls.

For a stratified sample of size n from strata of sizes N₁, N₂, …:

• Sample size from stratum i = (Nᵢ / total) × n.

Bias

A sample is biased if some members are more likely to be selected than others. Examples:

• Asking only Year 11 students about school dinners (excludes Years 7–10).
• Online survey reaches only those with internet access.
• Volunteer sampling (only enthusiastic people respond).

Bias makes the sample unrepresentative — conclusions drawn don't generalise to the population.

Inferring population properties

If a sample is fair and representative, you can scale up:

• Sample of 60 includes 18 left-handed students. Estimate left-handers in the school of 1500: (18/60) × 1500 = 450.
• Capture-recapture (Higher): tag M animals, release, later catch n animals of which m are tagged. Estimate population N ≈ Mn/m.

OCR mark scheme conventions

• "State a reason" → B1 for any single valid reason.
• "Comment on the validity" → expects identification of a flaw + explanation. Two B1 marks typical.
• For stratified sampling: M1 for correct method (Nᵢ/total × n), A1 for the value, possibly with rounding to a whole number.