Simple Random Sample in AP Statistics

In AP Statistics, a simple random sample (SRS) of size n is chosen so that every possible group of n individuals from the population has an equal chance of being the sample. It is the classic way to meet the random condition for confidence intervals and significance tests.

What is Simple Random Sample?

A simple random sample (SRS) of size n is a sample picked so that every possible set of n individuals is equally likely to be the one you get. Picture writing every name on an identical slip, dumping them in a hat, mixing well, and drawing n slips. Any combination of n names could come out, and none is favored. In practice you use a random number generator or a table of random digits instead of a hat.

The key detail is that the definition is about samples, not just individuals. It isn't enough that each person has the same chance of being picked. Every group of size n must have the same chance. That one detail is what separates an SRS from other random designs like stratified or cluster sampling. Those designs are still random, but they make some combinations of individuals impossible.

Why Simple Random Sample matters in AP® Statistics

An SRS is the sampling method that inference is built on. When you see "independent simple random samples" in a problem, that phrase is handing you the random condition you need to check before running any procedure. That shows up all over Units 3 and 4. You'll see it in constructing a confidence interval for the difference between two proportions (3.10), setting up a two-proportion test (3.12), setting up a chi-square test for homogeneity or independence (3.14), and the two-sample procedures for means (4.7, 4.9, 4.10). The big idea is that random selection lets you generalize from the sample to the population. Without it, a sample can be biased, and every interval or p-value you compute is built on sand. For a full walkthrough of any of those procedures, head to the topic study guides. This page focuses on the SRS piece that ties them all together.

How Simple Random Sample connects across the course

Random Condition for Two-Proportion Inference (Unit 3)

In topics 3.10 and 3.12, the first box to check is "two independent random samples (or a randomized experiment)." An SRS from each population is the cleanest way to check it. Think of the SRS as your permission slip to say the difference you see in the sample reflects a real difference in the populations.

Independent Groups in Two-Sample t Procedures (Unit 4)

Two-sample t intervals and tests (4.7, 4.9, 4.10) need two independent samples, meaning who lands in sample 1 has no effect on who lands in sample 2. Taking a separate SRS from each population gives you that. If the same people are measured twice, you have paired data and need a different procedure.

10% Condition (Units 3-4)

An SRS is drawn without replacement, so each pick slightly changes the pool for the next one. The 10% condition (n ≤ 10% of the population) tells you that change is small enough to ignore. That keeps the standard error formulas accurate. The SRS gets you randomness, and the 10% condition makes sure you can still treat the observations as independent.

Chi-Square Homogeneity vs. Independence (Unit 3)

In 3.14, how the sample was taken tells you which test to use. Separate SRSs from two or more populations, compared on one categorical variable, point to a test for homogeneity. One SRS from a single population, classified on two categorical variables, points to a test for independence. Same chi-square math, different sampling story.

Is Simple Random Sample on the AP® Statistics exam?

You'll rarely be asked to just define an SRS. Instead, you'll use the fact that a sample is an SRS. Here's what that looks like:

• Checking conditions. In any inference FRQ, you earn credit by stating the random condition and citing the context, for example "the problem states both samples are independent simple random samples." A vague "it's random" usually isn't enough. Tie it to the stem.
• Reading MCQ stems. Stems often open with something like "Suppose you have two independent simple random samples..." and then give summary stats. One Fiveable practice question gives Sample 1 with n = 40, mean 50, SD 8 and Sample 2 with n = 35, mean 45, SD 10. That phrasing is your cue that the random and independence conditions are met and you're headed toward a two-sample t procedure.
• Generalizing conclusions. Random sampling lets you extend results to the population. Random assignment lets you make cause-and-effect claims. Knowing which one an SRS gives you is a common scoring point.
• Authentic framing. The term appears in 2025 FRQ Q4, so you can expect the College Board to keep using it in free-response stems.

Simple Random Sample vs Stratified random sample

Both are random, but they aren't the same design. In a stratified random sample, you split the population into similar groups (strata) and take a separate SRS within each one. That guarantees every stratum is represented. It also means some combinations, like a sample drawn entirely from one stratum, can never happen. So not every possible sample of size n is equally likely, which means it is not an SRS. A good gut check is to ask whether any group of n people could have been chosen. If yes, it's an SRS. If the design forces certain groups in or out, it isn't.

Key things to remember about Simple Random Sample

• A simple random sample of size n gives every possible group of n individuals the same chance of being selected.

• Giving each individual an equal chance is not enough on its own to make a sample an SRS, because stratified and cluster samples can do that too.

• An SRS is the standard way to satisfy the random condition for confidence intervals and significance tests in Units 3 and 4.

• Two separate SRSs from two populations give you the independent groups needed for two-sample proportion and two-sample t procedures.

• Because an SRS is drawn without replacement, you should also check the 10% condition before using standard error formulas.

• Random sampling lets you generalize to the population, but only random assignment in an experiment supports a cause-and-effect conclusion.

Frequently asked questions about Simple Random Sample

What is a simple random sample in AP Stats?

It's a sample of size n chosen so that every possible group of n individuals has an equal chance of being selected. On the exam, it's the go-to way to meet the random condition for inference.

If every person has an equal chance of being picked, is it a simple random sample?

Not necessarily. A stratified sample with proportional allocation can give every individual the same chance, but it still isn't an SRS because some combinations of people can never be chosen. The test is whether every possible sample of size n is equally likely.

What's the difference between a simple random sample and a stratified random sample?

An SRS draws n individuals directly from the whole population. A stratified sample first splits the population into strata and then takes an SRS within each stratum. Stratifying guarantees representation from each group and can reduce variability, but the overall sample is not an SRS.

How do I check the random condition on an AP Stats FRQ?

Point to the context. For example, write "the problem states these are two independent simple random samples." Then check the 10% condition, since an SRS is drawn without replacement.

Is simple random sample on the AP Stats exam?

Yes. It appears in inference stems throughout Units 3 and 4, including two-proportion procedures, chi-square tests, and two-sample t procedures, and it is used in 2025 FRQ Q4.