Systematic Sample vs Cluster Sample

Put the population in some order, fix an interval $k$ (the sampling interval), draw a starting position at random from the first $k$ places, then take every $k$-th unit after it. That is the whole procedure, and the only randomness in it is the start.

A club has 500 members and you want 50, so $k = 500/50 = 10$. Draw a random integer from 1 to 10, say 7, and the sample is members 7, 17, 27, and so on up to 497. Two facts follow. Every member sits in exactly one of the 10 possible samples, so each has probability $1/10$ of being chosen, and because those 10 samples split the club into 10 groups of 50, their means average to the club mean exactly, so the method is unbiased for $\mu$ (mu) whatever order the list is in. But only 10 samples are possible, out of roughly $10^{69}$ subsets of size 50, and members 7 and 8 can never appear together, since any two selected members differ by a multiple of 10.

The sentence to correct runs the other way: "picking every 10th customer is not random, because the interval is fixed." The random start is the randomness, and it is enough to give every member a known, equal chance. What it does not give is the property that defines a simple random sample, which is about groups rather than individuals. Systematic sampling is a random sampling method and it is not an SRS.

The failure case is a list whose order repeats on a cycle matching $k$. Survey every 7th day from a random start in the first week and every day in the sample falls on the same weekday, so a Wednesday start gives an all-Wednesday sample. The center does not shift, but your one sample sees a single phase of the cycle and can sit far from the truth. When the ordering has nothing to do with the variable being measured, a systematic sample behaves much like an SRS and is far easier to run at a door or a checkout line.

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.