Cluster Sampling in AP Statistics

Building a cluster sample one step at a time

A cluster sample starts with groups that already exist, like schools, city blocks, classrooms, or plots of land, and picks whole groups at random instead of picking individuals. It's one of the designs in Unit 1 (Exploring One-Variable Data and Collecting Data), Topic 1.11 Random Sampling, where the bigger goal is getting data that fairly represents a population. A cluster sample comes together in four steps:

• Split the whole population into clusters. These are often natural groups that already exist, like schools, city blocks, classrooms, or plots of land.
• Number the clusters (for example, 1 to 50).
• Take an SRS of clusters, using a random number generator or by drawing numbered slips of paper. Ignore repeated numbers so no cluster gets picked twice.
• Measure every individual in each chosen cluster.

The last step is what makes it a cluster sample. If you randomly picked only some students inside each chosen school, you'd be doing something different, and an AP grader will notice. The random part happens once, at the level of the clusters.

Worked example: 5 schools out of 50

A district wants the average reading level of its elementary students. It has 50 schools, and each school has a mix of students much like the district as a whole.

To take a cluster sample, number the schools 1 to 50. Use a random number generator to get 5 different integers from 1 to 50, skipping any repeats. Give the reading test to every student in those 5 schools.

This design makes sense here because each school already looks like the district. Testing all students at 5 schools is far cheaper and easier than tracking down students scattered across all 50 buildings for an SRS. The district also doesn't need a full list of every student before it starts. It only needs a list of schools. The same reasoning fits a forester who splits a forest into 1,000 equal plots, randomly picks 50, and measures every tree in those plots. Travel stays manageable, and the plots spread across the forest represent it.

Clusters vs. strata: telling the two designs apart

The biggest mix-up in Topic 1.11 is swapping cluster and stratified sampling. Both start by splitting the population into groups. What happens next is where they differ.

In a stratified sample, the groups (strata) are built so the people inside each one are similar, like grouping by grade level. Then you take an SRS from every stratum and combine them. In a cluster sample, each cluster ideally contains a varied mix that mirrors the population. You randomly pick some clusters and take everyone in them.

A quick check works on multiple-choice descriptions. If the study samples some individuals from every group, it's stratified. If it samples all individuals from some groups, it's cluster. A related multiple-choice question gives you how a variable is spread across groups and asks which design fits. Groups that differ from each other but are uniform inside point to stratification. Groups that each resemble the whole population point to clusters.

Cluster sampling has a weak spot. If the clusters are uniform inside and different from one another (say, wealthy neighborhoods and low-income neighborhoods as clusters), picking only a few clusters could miss whole parts of the population. That would make your estimate swing a lot from sample to sample.

Describing and defending a cluster sample in a free response

Sampling design questions show up in Question 1 (Multi-Focus, Practices 1 and 2), which centers on collecting data. They also appear in Question 4 (Multi-Focus, Practices 2, 3, and 4), where data collection is combined with analysis. The two tasks match the two learning objectives. You identify the method from a description, or you justify why a method fits.

A strong description of a cluster sample names the clusters in context, explains how they're randomly selected (numbering plus a random number generator or slips of paper), and states that every individual in the selected clusters is measured. Leaving out "every individual" is a common way to lose credit, because without it the reader can't tell your design from a stratified one.

A strong justification connects the design to the specific situation. Point out that each cluster contains a representative mix, so the clusters are similar to each other. Then add the practical gain: lower cost or travel, or no need for a list of every individual. Saying "it's easier" with no context usually isn't enough. Explain what makes it easier in this particular study, such as visiting 5 schools instead of 50.