Cluster sample

Clusters are non-overlapping groups that together cover the population, and a good cluster holds the same mix the population does, which leaves the clusters resembling one another. The randomness acts on the clusters, not on people: take a simple random sample of clusters, then collect data from every individual inside the ones drawn. Each cluster is measured completely or not at all.

Use the same school of 1,200 students that the stratified plan uses, this time split into 60 advisory groups of 20 that each mix all four grades. Draw 4 advisories at random and survey all 20 students in each: $4 \times 20 = 80$ students out of 4 groups, with the other 56 advisories contributing nobody. Every student still has a $4/60 = 1/15$ chance of being in the sample, the same rate the stratified plan gives.

That equal chance is where the misreading starts. "Every student had the same chance of being picked, so this is a simple random sample." It is not. An SRS requires every set of 80 students to be equally likely, and here only $\binom{60}{4} = 487{,}635$ samples can occur at all. Two students in the same advisory are either both in or both out, and no sample ever spans five different advisories. Equal chance for each individual is a weaker condition than the SRS condition.

The price of clustering is variability, not bias. Selection is random, so the method has no systematic lean, but neighbors resemble neighbors, so 80 students from 4 advisories usually carry less information than 80 drawn one at a time. More clusters therefore beat bigger clusters at a fixed total, and clusters built to be alike inside, one advisory per grade for instance, are the worst case. What is a cluster sample works through that cost.

Cluster sampling is Unit 1, topic 1.11 Random Sampling. Read the two divisions off a scenario and the naming question answers itself.