Why Sample Natural Groups?
Suppose a health department wants to learn about students’ breakfast habits across a large district. Visiting a random selection of individual students may require contacting many schools. If the department instead randomly selects a few classrooms and surveys every student in those classrooms, the fieldwork may be simpler. This approach is one-stage cluster sampling: the groups are selected at random, and everyone in each selected group is included.
A cluster is a naturally occurring group of individuals, such as a classroom, clinic, or household. The individuals remain the units being described; the groups are the units selected. Cluster sampling is useful when groups are convenient to reach, but it does not automatically give more precise results than sampling individuals directly.
How the Design Works
Let there be \(M\) clusters in the population, with \(N_j\) individuals in cluster \(j\). The total number of individuals is \(N=\sum_{j=1}^{M}N_j\). Select \(k\) clusters by simple random sampling without replacement from the complete cluster list. Then survey every eligible individual in each selected cluster. The resulting number of surveyed individuals is the sum of the sizes of the selected clusters; if group sizes differ, this sample size can vary from one possible selection to another.
For the design to represent the stated target population, the cluster list should cover that population: each eligible individual should belong to exactly one listed cluster, with no omissions or duplicate membership. Cluster selection must be genuinely random, and the plan must include all members of selected clusters. If some members are not surveyed, the actual data collection is no longer a complete one-stage cluster sample as defined here.
The distinction from stratified sampling is the selection plan. In stratified sampling, a sample is drawn from every stratum. In one-stage cluster sampling, only some clusters are selected, and every member of each selected cluster is included. A group’s role depends on the design, not just its name: a set of classrooms could serve as clusters or as strata under different plans.
Worked Example: Surveying Selected Classrooms
A district has 18 classrooms with 25 students in each classroom, for a total of \(18(25)=450\) students. A researcher randomly selects 3 classrooms from the list and asks every student in those rooms whether they ate breakfast that morning. Of the 75 surveyed students, 51 answer Yes. Assume the list includes every classroom in the target population, the selected classrooms are chosen at random, and all students in those rooms respond.
This is one-stage cluster sampling: the 3 classrooms are selected, and all 25 students in each selected classroom are surveyed. The sample size is \(3(25)=75\). Each classroom has the same chance of selection, so a particular classroom’s selection probability is
Every student belongs to one classroom, and everyone in a selected classroom is included. Thus each student also has a \(1/6\) chance of selection. The observed sample proportion who ate breakfast is
The sample estimates that 68.0% of the district’s students ate breakfast that morning. Because all classrooms have the same size, the proportion among the surveyed students is also the proportion obtained by expanding the sampled number of Yes responses to the district total: \(6(51)=306\) estimated Yes responses among 450 students, and \(306/450=0.680\). This is an estimate from a hypothetical sample, not a claim about actual district students.
Selection Chances and Population Estimates
If \(k\) of \(M\) clusters are selected by simple random sampling without replacement, every cluster has selection probability \(k/M\). A person is included exactly when that person’s cluster is selected. Therefore, every individual has the same inclusion probability \(k/M\), even when cluster sizes differ.
Equal inclusion probabilities do not mean that the sample behaves like a simple random sample of individuals. People in a selected cluster are included together, and the total number of surveyed individuals can depend on which clusters are selected. When clusters have unequal sizes, the proportion among surveyed individuals is not in general the appropriate design-based estimate of the population proportion. If the population size \(N\) is known, one way to estimate a population total is to multiply the observed total in the selected clusters by \(M/k\), then divide by \(N\) to estimate a population proportion.
The reason the expansion estimates a total can be shown directly. Let \(I_j=1\) if cluster \(j\) is selected and \(I_j=0\) otherwise. Its selection probability is \(P(I_j=1)=k/M\), so its expected contribution to the expanded total is
Adding this equality over all clusters gives \(E(\widehat{T})=\sum_{j=1}^{M}Y_j\), the true population total. Dividing by a fixed known \(N\) shows that \(\widehat{T}/N\) estimates the population proportion. This justification relies on random cluster selection with the stated equal probability; it does not make nonresponse or an incomplete cluster list harmless.
Worked Example: Estimating a Clinic Population Proportion
A hypothetical health network has 12 clinics and 720 eligible patients in total. A researcher randomly selects 3 clinics and surveys every eligible patient in them. The selected clinics have 40, 70, and 80 eligible patients, respectively. Among them, 30, 35, and 56 patients report having received a particular vaccination. Assume all 12 clinics are on the frame, every eligible patient belongs to exactly one clinic, selection is random, and every patient in a selected clinic responds.
Each clinic, and therefore each patient, has probability \(3/12=0.25\) of inclusion. The selected sample contains \(40+70+80=190\) patients, including \(30+35+56=121\) who report vaccination. The cluster expansion factor is \(12/3=4\), so the estimated number of vaccinated patients in the network is
Using the known population size of 720, the estimated population proportion is
The estimate is 67.2% of eligible patients in the network. Simply using the fraction in the surveyed patients would give \(121/190\approx 0.637\), or 63.7%. In this sample, that differs from the expanded estimate because the selected clinics together contain 190 patients, whereas three of the 12 clinics do not necessarily represent exactly one-quarter of all 720 patients. The expansion estimates the total first and then expresses it as a proportion of the known population size.
Worked Example: Checking the Sample Size Before Fieldwork
A school system has 16 classrooms. A researcher will randomly select 4 classrooms and survey every student in them. The four selected rooms contain 22, 24, 28, and 30 students. Assume the full classroom list is correct, the selection is random, and all students in each chosen classroom can be surveyed.
The realized sample size is
A particular classroom has selection probability \(4/16=0.25\), and each student has the same 0.25 inclusion probability because every student is included if their classroom is selected. The sample size is 104, not a fixed number specified by the plan: another valid selection of four classrooms could yield a different total. If the researcher instead selected four classrooms and surveyed only a few students in each, the design would include a second selection stage and would not be one-stage cluster sampling.
Why the Choice of Clusters Matters
Cluster sampling can reduce travel, contact, and administrative costs. A researcher may be able to reach everyone in a few selected classrooms more easily than to reach the same number of students scattered across many schools. The trade-off is that people in the same cluster may resemble one another. For example, students in one classroom may share a teacher or schedule. When outcomes within clusters are similar, collecting more responses from the same few clusters may provide less information than spreading observations across more clusters.
This does not mean cluster sampling is automatically inaccurate, or that every set of clusters will be internally similar. It means that sample size alone does not describe how informative a cluster sample is. The number of selected clusters, their sizes, and how outcomes vary within and between clusters all matter. A cluster design is often chosen for practical reasons, but its estimates should still be interpreted as estimates from the particular random selection made.
Common Mistakes
- Calling any group-based sample a cluster sample. Selecting some classrooms and then only some students in each is not one-stage cluster sampling; every member of each selected cluster must be included.
- Confusing clusters with strata. Selecting some groups and surveying everyone in those groups is cluster sampling. Drawing a sample within every group is stratified sampling.
- Assuming the sample size is always fixed. If cluster sizes differ, selecting the same number of clusters can produce different numbers of individuals.
- Using the surveyed-person proportion without checking the design. With unequal cluster sizes, the raw proportion in the selected individuals need not equal the expansion estimate based on a known population size.
- Ignoring the cluster list. Random selection from a list that omits eligible groups cannot represent the omitted members through randomization alone.
Check Your Understanding
Try these questions without looking back.
- A county randomly selects 5 of its 25 clinics and surveys every eligible patient at those clinics. What is the cluster, and what makes this a one-stage design?
- A district has 30 classrooms and randomly selects 6. What is the inclusion probability for a student if every student in a selected classroom is surveyed?
- In a one-stage cluster sample, selected classrooms contain 20, 24, and 26 students. What is the realized sample size? Could another selection of three classrooms have a different sample size? Explain.
- A researcher selects 4 clinics at random but surveys only 10 patients at each selected clinic. Is this one-stage cluster sampling? Why or why not?
- Explain why selecting some groups and surveying everyone in them is different from stratified sampling.