Simple Random Sample vs Cluster Sample

A simple random sample of size $n$, usually shortened to SRS, comes from a method under which every possible sample of $n$ individuals is equally likely to be the one drawn. The condition is about whole samples, not about individuals. Equal chance for each individual follows from it but does not imply it, and that gap is where most naming mistakes live.

With 5 students and $n = 2$ there are 10 possible samples, each of probability $1/10$, and each student sits in 4 of the 10, so every individual has chance $4/10 = 2/5$. Now take a different plan for four students A, B, C and D: flip one fair coin, take A and B on heads, take C and D on tails. Every student still has chance $1/2$, and the sample size is still 2, but only 2 of the 6 possible pairs can ever come up. The pair A and C has probability 0. The plan uses chance, and it is not an SRS.

So distrust the sentence "everyone had an equal chance of being picked, so it was a simple random sample." Equal individual chance is necessary and not sufficient. A cluster sample gives every individual the same chance while making most subsets impossible, which is why the SRS condition is written in terms of samples.

Carrying one out needs a list. Number a roster of 500 students 1 to 500, generate random integers in that range, ignore repeats, and take the first 30 distinct labels. If the roster leaves people out, the result is an SRS of the sampling frame and still biased for the population, because no amount of randomness inside a short list reaches anyone who is not on it.

Random selection is topic 1.11 Random Sampling in Unit 1. It is what supports generalizing to the population; random assignment is the separate move that supports a cause-and-effect claim.

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.