---
title: "Independent Random Samples — AP Stats Definition & Guide"
description: "Independent random samples are two separately drawn random samples where one selection doesn't affect the other. They unlock every two-sample inference procedure in AP Stats Units 6-8."
canonical: "https://fiveable.me/ap-stats/key-terms/independent-random-samples"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 4"
---

# Independent Random Samples — AP Stats Definition & Guide

## Definition

Independent random samples are two samples selected randomly from two different populations, where choosing one sample has no effect on choosing the other. On the AP Stats exam, this is the independence condition you must verify before any two-sample z- or t-procedure for proportions or means.

## What It Is

Independent random samples are exactly what the name says. You take a [random sample](/ap-stats/key-terms/random-sample "fv-autolink") from population 1, you take a separate random sample from population 2, and neither selection influences the other. Sampling 120 adults from City A and 150 adults from City B is independent [random sampling](/ap-stats/unit-6/justifying-claim-based-on-confidence-interval-for-difference-population-proportions/study-guide/m6d2wvAbSyMqmNwEyt43 "fv-autolink"). Measuring the same 50 people before and after a treatment is not, because the two data sets share the same individuals.

In the [AP Statistics](/ap-stats "fv-autolink") CED, this phrase shows up over and over in the conditions for two-sample inference. For a two-sample z-interval or z-test for a difference of proportions (Topics 6.8 and 6.10), and a two-sample t-interval or t-test for a difference of means (Topics 7.6, 7.8, and 7.9), the independence condition reads the same way every time. Data should be collected using two independent, random samples or a randomized experiment. That last clause matters. Random assignment in an experiment creates the same kind of independence between groups, so an experiment can satisfy the condition even without random sampling from populations.

## Why It Matters

This term is the gatekeeper for every two-sample procedure in Units 6 and 7. Learning objectives 6.8.B, 6.10.C, 7.6.B, and 7.8.C all require you to verify [independence](/ap-stats/key-terms/independence "fv-autolink") by stating that the data came from two independent random samples or a [randomized experiment](/ap-stats/key-terms/randomized-experiment "fv-autolink"), plus the 10% condition when sampling without replacement. If the samples aren't independent (for example, paired before-and-after data), the entire procedure changes. You'd use a one-sample t-procedure on the differences instead.

It also matters because the [standard error](/ap-stats/key-terms/standard-error "fv-autolink") formulas for p̂₁ - p̂₂ and x̄₁ - x̄₂ assume independence. The formula √(s₁²/n₁ + s₂²/n₂) adds the two variances, and that addition only works when the samples don't influence each other. So this isn't just a box to check on an FRQ. It's the mathematical foundation that makes the formulas on your formula sheet valid.

## Connections

### Dependent (Paired) Samples (Unit 7)

This is the flip side and the most tested contrast. If the same individuals appear in both data sets, or [subjects](/ap-stats/unit-3/intro-experimental-design/study-guide/gsdVWumN3cEYmXOIVv95 "fv-autolink") are matched in pairs, the samples are dependent and you analyze the differences with a one-sample t-procedure. Choosing two-sample versus paired comes down to one question: are these two separate groups, or one group measured twice?

### [10% Condition (Unit 5-7)](/ap-stats/key-terms/10percent-condition)

Independence has two layers in the conditions check. Independent random samples handle independence between the two groups, while the [10% condition](/ap-stats/key-terms/10percent-condition "fv-autolink") (n₁ ≤ 10% of N₁ and n₂ ≤ 10% of N₂) handles independence within each sample when you sample without replacement. On an FRQ, you usually need to address both.

### Chi-Square Test for Homogeneity (Unit 8)

The homogeneity test is basically the two-proportion comparison extended to more groups or more categories, and it inherits the same data-collection requirement. Topic 8.5 asks for independent random samples from each population (a stratified random sample) or a randomized experiment. The 2021 FRQ used three independent random samples of teens for exactly this setup.

### Sampling Distribution of a Difference (Units 6-7)

Independence is what lets you add variances. The standard deviation of x̄₁ - x̄₂ is √(σ₁²/n₁ + σ₂²/n₂) only because the two samples don't co-vary. If the samples were dependent, this formula would be wrong, which is the deep reason the exam makes you verify this condition.

## On the AP Exam

Multiple choice questions love to hand you a scenario like "independent random samples of 120 adults from City A and 150 from City B" and ask which conditions must be checked, or whether a two-sample or paired procedure is appropriate. You should be able to spot independent samples in a stem and match them to the right procedure (two-sample z for proportions, two-sample t for means).

On FRQs, this phrase appears in the inference rubric. The 2026 FRQ Q1 gave independent random samples of 14 goats from each of two breeds, and the 2021 FRQ Q5 used three independent random samples of teens for a chi-square setting. When you carry out a two-sample procedure, you earn condition-checking credit by explicitly writing that the data came from two independent random samples (or a randomized experiment) and verifying the 10% condition for each sample. Just writing "random ✓" without naming independence between the samples can cost you.

## independent random samples vs Dependent (paired) samples

Independent random samples come from two separate groups with no link between them, so you use two-sample procedures and add variances in the standard error. Dependent samples involve the same subjects measured twice or matched pairs, so you compute a difference for each pair and run a one-sample t-procedure on those differences. Quick test: if you can pair each value in sample 1 with a specific value in sample 2 (same person, same twin pair, same plot of land), the samples are dependent. If picking sample 1 tells you nothing about who's in sample 2, they're independent.

## Key Takeaways

- Independent random samples means two separately drawn random samples where selecting one has no effect on selecting the other.
- Every two-sample inference procedure in Units 6 and 7 requires data from two independent random samples or a randomized experiment as part of the independence condition.
- A randomized experiment satisfies the same condition because random assignment makes the treatment groups independent.
- When sampling without replacement, pair this condition with the 10% condition for each sample (n₁ ≤ 10% of N₁ and n₂ ≤ 10% of N₂).
- If the data are paired (same subjects measured twice or matched pairs), the samples are dependent and you must use a one-sample t-procedure on the differences instead.
- Independence is what justifies adding variances in the standard error formulas like √(s₁²/n₁ + s₂²/n₂).

## FAQs

### What are independent random samples in AP Stats?

They're two samples selected randomly from two different populations, where the selection of one sample doesn't influence the other. They're required (or a randomized experiment is) before you can run any two-sample z- or t-procedure.

### How are independent samples different from paired (dependent) samples?

Independent samples are two unrelated groups, like 14 goats from Breed H and 14 from Breed J. Paired samples link each observation in one set to a specific observation in the other, like the same students tested before and after a course. Independent data gets a two-sample procedure; paired data gets a one-sample t-test on the differences.

### Do I always need independent random samples for a two-sample test?

No. The CED gives two routes to independence: two independent random samples OR a randomized experiment. Random assignment to treatment groups in an experiment satisfies the condition even without random sampling.

### Is checking 'random' the same as checking 'independent random samples'?

Not quite. You need to state that both samples were randomly selected AND that they're independent of each other, then check the 10% condition for each sample if sampling without replacement. Skipping the independence-between-samples part can lose condition-checking credit on an FRQ.

### Do independent random samples need to be the same size?

No. The samples can have different sizes, like 250 households in one county and 300 in another. The standard error formula handles unequal sizes since each sample's variance is divided by its own n.

## Related Study Guides

- [4.7 Constructing a Confidence Interval for the Difference Between Two Population Means](/ap-stats/unit-4/confidence-intervals-for-difference-two-means/study-guide/4eW27Kpyfs2J6HUNvehc)

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