Why does sampling whole classrooms instead of individual students cost you effective sample size?

## Why anyone clusters in the first place Cluster sampling selects naturally occurring groups — schools, classrooms, city blocks, households, stores — and measures units inside the selected groups. **Multistage** sampling nests the idea: sample schools (primary sampling units), then classrooms inside the selected schools (secondary units), then measure every student in the selected classrooms, or a subsample of them. Two reasons drive it, and neither is statistical elegance. **Cost.** Measuring 1,000 students scattered across a country means 1,000 travel legs. Measuring 40 classrooms of 25 means 40 site visits. The fieldwork saving is enormous. **Frames.** You often cannot draw a simple random sample of students because no national list of students exists — but a list of schools does. Multistage designs let you build the frame one stage at a time: you only need to enumerate students inside the schools you actually selected. ## The cost: correlated observations Units inside a cluster are alike. Students in one classroom share a teacher, a curriculum, a catchment area and a peer group; households on one block share income level and infrastructure. The **intraclass correlation** `rho` quantifies this: it is the correlation between two units drawn from the same cluster, equivalently the share of total variance that sits *between* clusters rather than within them. `rho = 0` means cluster membership tells you nothing; `rho = 1` means every unit inside a cluster is identical, so the second student measured in a classroom adds literally nothing. This is exactly the opposite of what stratification exploits. Strata are built to be internally *homogeneous* and are *all* sampled, which helps. Clusters are internally homogeneous and only *some* are sampled, which hurts. The same variable — school — can play either role: sample every school and it is a stratum, sample 40 of 2,000 and it is a cluster. ## The design effect The **design effect** is the ratio of the variance your design actually produces to the variance a simple random sample of the same size would have produced: `Deff = Var(design) / Var(SRS, same n)`. For a cluster design with equal cluster size `m` and intraclass correlation `rho`, `Deff is approximately 1 + (m - 1) * rho` Read the formula. When `rho = 0`, `Deff = 1` and clustering is free. When `m = 1` — one unit per cluster, so you sampled units individually after all — `Deff = 1` again. Everywhere else `Deff` grows with *both* cluster size and correlation, and `m` enters multiplied by `rho`, which is why even a tiny `rho` becomes serious once clusters are large. The practical translation is **effective sample size**: `n_eff = n / Deff`, the number of independently drawn observations that would carry the same information. Work the classroom case. `m = 25`, `rho = 0.1`: `Deff = 1 + 24 * 0.1 = 3.4`. Measure 1,000 students and you hold roughly `1000 / 3.4`, about 294 independent students' worth of information. Two thirds of the fieldwork has bought you nothing in precision — and it may still have been the right call, because 1,000 clustered measurements can be cheaper than 294 scattered ones. ## Fewer per cluster, more clusters The formula immediately implies the design rule. Halve `m` from 25 to 10 at the same `rho = 0.1` and `Deff` falls from 3.4 to 1.9. With a fixed total of 400 students, 40 classrooms of 10 give a far more precise mean than 10 classrooms of 40. Precision is driven mainly by the **number of clusters**, because clusters are the units that vary independently. Against this pushes cost. Each additional site has a fixed cost — travel, permission, setup — while each additional student inside an already-visited classroom is nearly free. The optimal cluster size balances that cost ratio against `rho`: the higher the intraclass correlation and the cheaper the extra site, the smaller the clusters should be. ## Beyond the simple formula The `1 + (m - 1) * rho` expression assumes equal cluster sizes. Unequal cluster sizes inflate the design effect further, and so do unequal sampling weights — those contributions are usually folded into a single reported `Deff` for the survey. A design effect below 1 is possible and means the design beat simple random sampling: stratification typically produces `Deff` slightly under 1, and a well-ordered systematic draw can too. Cluster designs essentially always produce `Deff` above 1, because `rho` in real social, educational and geographic data is small but reliably positive. ## Where candidates lose the point The fatal move is to plan or report as if the 1,000 clustered students were 1,000 independent observations. Every precision statement, every sample-size calculation and every test built on that assumption is overconfident by a factor of `sqrt(Deff)` in the standard-error scale. The correct planning move is the reverse: decide the effective sample size the decision requires, guess `rho` from prior studies of similar populations, choose `m` from cost, and inflate the nominal sample size by `Deff`. Interviewers ask this question because it separates people who have designed field data collection from people who have only analysed clean rectangles of independent rows.