Words to know
simple random sample (SRS)
A sample chosen so that every possible group of n individuals has the same chance of being the sample.
systematic sample
Choose a random starting point, then select every kth individual from a list, where k = N ÷ n is the sampling interval.
stratified sample
Divide the population into groups (strata) that share a characteristic, then take a random sample from each stratum, usually in proportion to its size.
cluster sample
Divide the population into groups (clusters), randomly choose some clusters, and include every individual in the chosen clusters.
multistage sample
A sample chosen in stages, with random selection at each stage (for example: random schools, then random classes, then random students).
convenience sample
A sample of individuals who are easy to reach. It is non-random and often biased.
voluntary-response sample
A sample made of people who choose to respond, such as an online poll or call-in show. It is non-random and usually biased.
sampling with / without replacement
With replacement, a chosen individual can be chosen again; without replacement, each individual can be chosen at most once. Surveys sample without replacement.
Stirring the soup properly
Last lesson you learned that a good sample must be random. But "random" can be done in several ways, each suited to a different situation. A polling company, a factory, a wildlife biologist and Statistics Canada all use random sampling, yet they choose their samples in quite different ways.
In this lesson you will learn five random (probability) sampling methods and two non-random methods that you should recognize and usually avoid.
1. Simple random sample (SRS)
Every individual, and every possible group of n individuals, has the same chance of being selected. To take an SRS:
- Number every member of the sampling frame from 1 to N.
- Generate random numbers in that range with a random-number table, the RanInt or Ran# key on many scientific calculators, or a spreadsheet formula such as
=RANDBETWEEN(1, 840). - Skip numbers outside the range and repeats, until you have n different individuals.
Strength: simple and unbiased. Weakness: you need a complete list, and by chance an SRS can under-represent a small group.
2. Systematic sample
Compute the sampling interval k = N ÷ n (round down to a whole number if needed). Choose a random start between 1 and k, then take every kth individual.
Strength: quick and easy, even without a full list (every 10th customer through a door). Weakness: biased if the list has a repeating pattern that lines up with k (for example, every 7th day of a list of dates is always the same weekday).
3. Stratified sample
Split the population into strata: groups whose members share a characteristic that might affect the results (grade, province, age group). Take a separate random sample from each stratum. In a proportional stratified sample, each stratum's sample size matches its share of the population:
stratum sample size = (stratum size ÷ population size) × total sample size.
Strength: guarantees every group is represented and often gives more precise results. Weakness: you must know each individual's stratum in advance.
4. Cluster sample
Split the population into clusters, usually natural groups such as homerooms, city blocks or boxes of product. Randomly choose some clusters and include everyone in them.
Strength: cheap and practical when individuals are spread out; no complete list of individuals is needed. Weakness: if clusters differ from one another, the result depends heavily on which clusters were chosen, so precision is lower.
5. Multistage sample
Select randomly in stages: randomly choose school boards, then schools within them, then classes, then students. Large national surveys use multistage designs because they are practical across a huge country.
Strata and clusters are easy to confuse. Stratify to make sure every group is in the sample (sample from every stratum). Cluster to save effort (take all of some clusters). Ideally, strata are different from each other and similar inside; clusters are similar to each other and mixed inside, like mini-populations.
Hands-on activitySampling a deck of cards
You need: a standard 52-card deck (jokers removed), paper and a pencil.
The question: what proportion of the deck are face cards (jacks, queens, kings)? You know the answer is 12/52 ≈ 0.231, so you can judge each method.
- SRS: shuffle well and deal 8 cards. Record the proportion of face cards.
- Systematic: shuffle, then lay the deck face down. Roll a die (or pick a random number from 1 to 6) for a start s, and take cards s, s + 6, s + 12, ..., for 8 cards.
- Stratified: sort the deck into the four suits (strata). Shuffle each suit and deal 2 cards from each: 8 cards.
- Cluster: sort the deck into 13 clusters by rank (all four aces, all four 2s, ...). Randomly choose 2 ranks and take all 4 cards of each: 8 cards.
- Repeat each method three times. Make a table of the 12 proportions.
What to look for: SRS, systematic and stratified results scatter around 0.23. The cluster results jump between 0, 0.5 and 1, because each rank cluster is all face cards or none. Clusters that are alike inside make cluster sampling very imprecise; clusters should be mixed, like mini-decks.