Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A random sample uses a chance process (random number generator, well-mixed numbered slips) to choose every individual.
- Without replacement: a chosen individual is not returned, so it can be chosen only once. With replacement: it is returned and could be chosen again.
- Surveys of people normally sample without replacement: ignore repeated labels.
- In an SRS of size n, every possible sample of size n is equally likely.
Key relationships
| Method | How it works | Good when… |
|---|---|---|
| Simple random sample | label 1 to N; choose n different labels by chance | a full list exists and no groups matter |
| Stratified | split into strata of similar individuals; SRS within every stratum; combine | groups differ on the variable, or results are needed for each group |
| Cluster | split into clusters that each mirror the population; SRS of clusters; measure everyone in them | the population is spread out and visits are costly |
| Systematic | k ≈ N ÷ n; random start from 1 to k; then every k-th individual | individuals come in a line or a long list with no repeating pattern |
Assumptions and conventions
- With a line of random digits, labels all have the same number of digits (001 to 250, not 1 to 250).
- Skip numbers that are not labels, and skip repeats unless sampling with replacement.
- Stratified and systematic samples are random samples, but they are not SRSs: some samples of size n cannot happen.
- Proportional allocation in strata is common, not required.
Mistakes to avoid
- Calling a haphazard or convenience choice “random”.
- Saying “each individual has the same chance” as the definition of an SRS. It is every sample of size n.
- Swapping strata and clusters.
- Sampling a few people from every cluster. In a cluster sample you take everyone in the chosen clusters.
- Forgetting the random start in a systematic sample.
- Stratifying by a characteristic that has nothing to do with the variable measured.
- A justification with no context: name the population, the groups and the variable.
Quick self-check
- A list has 500 names and you want a systematic sample of 25. What is k, and which names are chosen if the random start is 13? (k = 500 ÷ 25 = 20; names 13, 33, 53, … , 493)
- A school has 300 boarders and 200 day students. Take a stratified sample of 50 in proportion to size. How many from each group? (30 boarders and 20 day students)
- A pollster picks 4 blocks of flats at random and interviews every resident. Name the method. (Cluster random sample, with blocks of flats as clusters)
Next: practice questions.