AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/1/1-11)
Unit 1 · Topic 1.11
1.11 Random Sampling
Random sampling uses chance, not personal choice, to pick who's in the sample. You'll learn four methods (simple random, stratified, cluster and systematic), how to carry each one out and how to explain why one fits a situation better than another.
Key terms
- simple random sample (SRS)
- stratified random sample
- cluster sample
- systematic random sample
- sampling with / without replacement
With or without replacement
Sampling without replacement means a unit can be chosen only once. Once a name is drawn, it stays out. Sampling with replacement puts each chosen unit back, so it could be picked again. Almost all real surveys sample without replacement.
Simple random sample (SRS)
In an SRS of size n, every possible group of n units has the same chance of being the sample. To take one, give every unit in the population a number from 1 to N, use a random number generator to pick n different numbers (ignoring repeats), and select those units. Drawing numbered slips from a well-mixed hat also works.
Describe the process clearly enough that someone else could do it: what's numbered, how numbers are generated and what to do with repeats.
Stratified random sample
Split the population into groups called strata, where units in each stratum are similar to each other in some way that likely affects the response (like grade level or region). Take an SRS within each stratum and combine them.
Stratifying guarantees every group is represented. When the strata really differ on the variable you're measuring, it also gives more precise estimates than an SRS of the same size.
Cluster sample
Split the population into groups called clusters, usually based on location, like homerooms or city blocks. Ideally each cluster is a mini-version of the population, with a mix of different kinds of units. Randomly choose some clusters and collect data from every unit in the chosen ones.
Cluster sampling saves time and money because you only travel to a few places. Don't confuse it with stratified: stratified takes some from every group; cluster takes all from some groups.
Systematic random sample
Choose a random starting point, then take every kth unit from a list or a line. For example, pick a random number from 1 to 20 and survey that customer and every 20th customer after. It's easy to carry out, but it can go wrong if the list has a repeating pattern that lines up with k.
| Method | How it works | Main advantage |
|---|---|---|
| SRS | Every group of n equally likely | Simple and unbiased |
| Stratified | SRS from each similar group | Every group represented; more precise |
| Cluster | Randomly pick groups, take everyone in them | Cheaper, easier to reach |
| Systematic | Random start, then every kth | Easy with a list or a line |
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Describe an SRS
A school has 1,200 students. Describe how to select a simple random sample of 60 students.
Show the solutionHide the solution
- Step 1: Get a list of all 1,200 students and number them 1 to 1,200.
- Step 2: Use a random number generator to produce integers from 1 to 1,200.
- Step 3: Ignore repeats and keep going until you have 60 different numbers.
- Step 4: The students with those numbers are the sample.
Answer: Number the students 1–1,200, generate random integers from 1 to 1,200 until you have 60 different ones (skip repeats), and survey those 60 students.
- Example 2Calculator allowed
Choose stratified or cluster
A district wants to estimate the average number of hours students in grades 9–12 spend on homework per week. Homework time is known to increase from 9th to 12th grade. Should the district stratify by grade or use grade levels as clusters? Explain.
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- Step 1: Students within the same grade are similar in homework time, and grades differ from each other. That's the setup for strata, not clusters.
- Step 2: Stratify: take an SRS from each grade, for example 50 students per grade, and combine them.
- Step 3: This guarantees all four grades are represented and reduces variability in the estimate, because each stratum's homework times are similar.
- Step 4: Using grades as clusters would mean surveying every student in one or two randomly chosen grades, and the result could badly misrepresent the district if, say, only 9th grade were picked.
Answer: Stratify by grade. Homework time is similar within each grade but differs between grades, so taking an SRS from each grade gives a more precise estimate than an SRS or a cluster sample of grades.
Common mistakes
- Mixing up stratified and cluster. Stratified: sample from every group. Cluster: take everyone in a few groups.
- Describing a sample as random without saying how chance is used ("pick 60 students at random" isn't a full description).
- Forgetting to deal with repeated random numbers when sampling without replacement.
- Thinking that a random sample guarantees a perfectly representative sample. It avoids bias but still varies by chance.
On the exam
- Expect to describe how to carry out a sampling method in enough detail that someone else could follow it, including numbering, the random mechanism and repeats.
- "Why stratify?" answers should mention that the response differs between the strata and is similar within them, in context.
Connected topics
Videos
Check yourself
4 questions on 1.11 Random Sampling. Pick an answer to see if you got it, and why.
A university wants to survey 200 of its students. It randomly selects 50 first-year, 50 second-year, 50 third-year and 50 fourth-year students. What type of sample is this?
A city wants to estimate the average age of its trees. It divides a map of the city into 120 blocks, randomly selects 8 blocks, and records the age of every tree on those 8 blocks. What type of sample is this?
A teacher has 15 boys and 15 girls in her class. She puts the boys' names in one hat and the girls' names in another, then draws 3 names from each hat. Why is this NOT a simple random sample of 6 students?
A store wants to survey about 5% of the 2,000 customers expected on Saturday. Which plan describes a systematic random sample?
0 of 4 answered