Cluster Sample vs Stratified Random 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.

Strata are non-overlapping groups that together cover the whole population, built so the individuals inside one stratum are alike on a trait related to what is being measured, and unlike the individuals in the other strata. The randomness then acts inside every stratum: a separate simple random sample is drawn from each one and the pieces are combined into a single sample. No stratum can be missed, because the design reaches into all of them.

Take 1,200 students at one school, 300 in each of grades 9 through 12, with a plan that draws an SRS of 20 students per grade. The sample is $4 \times 20 = 80$ students and every grade is guaranteed 20 seats in it. Each student's chance of selection is $20/300 = 1/15$, which matches the overall rate $80/1200$. Equal rates across strata is called proportional allocation, and it is a choice rather than part of the definition; strata of unequal size and unequal sampling rates are still strata.

Here is the sentence to catch: "I randomly picked 4 of the school's advisory groups and surveyed everyone in them, so the sample is stratified." Dividing the population first is not what makes a design stratified. What makes it stratified is that the randomness selects individuals within every group. Randomness that selects whole groups, with everyone inside the chosen ones measured, is a cluster sample. Some of every group is stratified, all of a few groups is cluster.

Stratifying pays off only when the strata differ from one another on what you are measuring. If ninth and twelfth graders answer the question about the same way, stratifying by grade leaves you with an estimate about as variable as an SRS of 80, so it costs nothing and buys nothing. The gain comes from taking the between-stratum differences out of the sampling error.

Strata are not blocks. You stratify a population before selecting a sample, while blocking groups experimental units before assigning treatments. Sampling methods are Unit 1, topic 1.11 Random Sampling.