Two Ways to Use Groups in a Random Sample
In Stratified Random Sampling, the population is divided into groups, and individuals are selected at random from every group. In Cluster Sampling, some groups are selected at random, and every individual in each selected group is included. Both designs use chance, but they select different things.
The distinction matters when planning a study. If a school wants to hear from students in every grade, stratified sampling can ensure that each grade is represented by students chosen at random. If it is easier to survey whole homerooms than to contact selected students across the school, cluster sampling may be more practical. The group labels alone do not determine the design; what matters is how selection happens.
A stratum is one of the nonoverlapping groups used in a stratified design. A cluster is one of the nonoverlapping groups used in a cluster design. In either design, the groups should together cover the population, and each individual should belong to exactly one group.
One School, Two Sampling Plans
Suppose Northview High School has 600 students: 150 in each of grades 9, 10, 11, and 12. The school wants to ask students how many days per week they use the library. For a stratified design, it could treat each grade as a stratum and randomly select 15 students from each grade, for 60 students total.
For a cluster design, suppose the school has 20 homerooms of 30 students each. It could randomly select two homerooms and survey all 30 students in each selected homeroom, also for a total of 60 students. The number of students happens to match the stratified plan, but the process is different: one selects individual students from every grade; the other selects whole homerooms and includes everyone in those homerooms.
This example assumes each student appears on the school’s list and is assigned to one grade and one homeroom. A suitable sampling frame matters, as discussed in Population, Sampling Frame, and Sample. Random selection can help limit deliberate selection, as explained in Why Random Selection Matters, but it does not correct an incomplete list or ensure that a selected person responds.
Key Differences at a Glance
| Feature | Stratified random sampling | Cluster sampling |
|---|---|---|
| How the population is divided | Into strata, such as the four grade levels | Into clusters, such as the 20 homerooms |
| What is selected at random | Individuals from every stratum | Some whole clusters |
| Who is included | The selected individuals from each stratum | Every individual in each selected cluster |
| Are all groups represented? | Yes: individuals are selected from every stratum | Not necessarily: only selected clusters are included |
| Possible reason to use it | To ensure the sample includes individuals from each important group | To make data collection more practical when groups are convenient to reach |
| School example | Randomly select 15 students from each grade | Randomly select two homerooms and survey all students in them |
The sample sizes in the table are not rules. A stratified design does not require selecting the same number from every stratum. A researcher could select different numbers from different grades, depending on the study’s purpose and plan. What makes it stratified is selecting individuals at random from every stratum. Likewise, a cluster design does not require selecting exactly two clusters; it requires randomly selecting clusters and including all members of each selected cluster.
Worked Example: Choosing a Design for a School Survey
Worked Example: Hearing From Every Grade
Northview’s student council wants to estimate how many days per week students use the library. It particularly wants to make sure the sample includes students from all four grades. The school has a list of all 600 students, grouped by grade, with 150 students in each grade.
State: The population is all 600 Northview students. The observational unit is one student, and the variable is the number of days per week that student uses the library.
Plan: Use stratified random sampling, with grade as the stratum variable. Divide the list into the four nonoverlapping grade groups, then use a chance method to select 15 students from each grade. This meets the council’s aim of including randomly selected students from every grade.
Do: The sample contains \(4 \times 15=60\) students. The council contacts the selected students from grades 9, 10, 11, and 12 and asks each the same clearly worded question. If a selected student does not respond, the council should record the nonresponse rather than quietly replace that student with a convenient volunteer.
Conclude: This is stratified random sampling because individuals are selected at random from every grade. It is not cluster sampling: the council does not select certain grades and survey every student in them. The plan directly supports having students from all four grades represented in the sample, although random selection does not guarantee that the sample will perfectly match every feature of the school.
Why Choose One Design Over the Other?
Stratified sampling is useful when the study needs representation from each of several important groups. For example, grade may matter because library use could differ among grade levels. Selecting students from every grade prevents the design from accidentally leaving a grade out. Within each grade, chance determines which individuals are selected.
The number chosen from each stratum should be part of the plan, not an assumption that every group must contribute the same number. In the example, each grade has 150 students, so selecting 15 from each gives equal-sized samples from equally sized grades. If the grade populations were different in size, the council might choose sample sizes that reflect those differences or use another allocation that suits its goal. The sampling method is still stratified as long as individuals are selected from every stratum.
Cluster sampling may be useful when groups are already organized and it is easier to survey everyone in a few selected groups than to locate individuals spread throughout the school. Surveying whole homerooms, for example, might let a teacher distribute and collect questionnaires during class. The trade-off is that some homerooms are left out entirely, and students in a selected homeroom are all included.
The group structure can influence how well a design serves its goal. If library use differs substantially from homeroom to homeroom, selecting just a few homerooms might give a sample whose responses depend heavily on which homerooms happen to be chosen. If each homeroom contains students with a range of library-use habits, the selected homerooms may provide a broader mix. These are considerations when designing a study, not guarantees about what a random sample will show.
Worked Example: Making the Homeroom Plan Explicit
Worked Example: Surveying Selected Homerooms
Northview’s library staff also want student feedback about the location of a new book-return bin. To keep data collection manageable, they plan to survey students in two homerooms. There are 20 homerooms, each with 30 students.
State: The population is all 600 Northview students. The observational unit is one student, and the variable is the student’s preference about the proposed bin location.
Plan: Use cluster sampling. Treat each homeroom as a cluster, label the 20 homerooms, and use a chance method to select two different homerooms. Selecting two clusters gives every homeroom a chance to be chosen, rather than letting staff choose the rooms they expect to be easiest.
Do: Survey all 30 students in each selected homeroom. The total sample size is \(2 \times 30=60\) students. For example, if homerooms 4 and 17 are selected, survey every student in homeroom 4 and every student in homeroom 17.
Conclude: This is cluster sampling because the randomly selected units are whole homerooms and every student in those selected homerooms is included. It would become stratified sampling if staff instead selected individual students at random from every homeroom. The two plans may produce the same sample size, but they do not select the sample in the same way.
Worked Example: Classifying a Proposed Plan
Worked Example: Is It Stratified or Cluster Sampling?
A counselor wants to learn how students travel to school. The counselor divides the student list by grade, then randomly selects 10 students from grade 9, 10 from grade 10, 10 from grade 11, and 10 from grade 12. The counselor surveys only those selected students.
State: The population is Northview’s 600 students. The groups in the plan are grade levels, and the sample consists of 40 students.
Plan: Classify the design by asking two questions: What units are chosen at random, and are selections made from every group? Here, individual students are selected from every grade.
Do: The plan selects 10 individuals from each of four strata, so it is stratified random sampling. It does not select some grade levels as whole clusters; nor does it survey every student in any selected grade.
Conclude: The counselor’s plan is stratified random sampling. To make it a cluster design using grades, the counselor would randomly select some grades and survey every student in each selected grade. That alternative could create a much larger sample, since each grade has 150 students.
Common Mistakes and AP Exam Tips
- Thinking any use of groups means cluster sampling. Both designs divide a population into groups. Name what is randomly selected: individuals from every group means stratified; some whole groups means cluster.
- Saying a stratified sample surveys everyone in a selected group. It surveys the individuals selected from each stratum, not every member of those strata.
- Saying a cluster sample selects a few people from every group. That describes stratified sampling. In cluster sampling, every member of each selected cluster is included.
- Assuming every stratum must contribute the same number. Equal numbers are one possible plan, not part of the definition. State the actual selection plan.
- Assuming every cluster is represented. A cluster sample uses only the clusters randomly selected, so some clusters are not included in the sample.
- Confusing a reason to choose a design with a guarantee. Stratification can ensure that the sample includes individuals from each stratum; it does not guarantee that every characteristic will be represented perfectly. Cluster sampling can make data collection practical; it does not guarantee that the selected clusters mirror all other clusters.
- Leaving out what happens after selection. A full description says whether selected individuals alone are surveyed or every person in selected groups is surveyed.
For a clear answer, identify the population, name the groups, state what is selected at random, and explain who is included in the sample. In context, write something like: “The school randomly selects students from each grade and surveys only those students, so this is stratified random sampling.” Or: “The school randomly selects homerooms and surveys every student in those homerooms, so this is cluster sampling.”
Check Your Understanding
Use the selection method—not just the group names—to classify each plan.
- A school randomly selects 12 students from each grade and surveys only those students. Is this stratified or cluster sampling? Explain.
- A school randomly selects three homerooms and surveys every student in those rooms. Is this stratified or cluster sampling? Explain.
- Does a stratified design require selecting the same number of students from every stratum?
- In cluster sampling, what happens to students in clusters that were not selected?
- Explain why two plans with the same number of students might still use different sampling methods.