---
title: "Dependent Samples — AP Stats Definition & Exam Guide"
description: "Dependent samples are paired or matched observations, like before-and-after scores. They decide whether you run paired t procedures or two-sample t in Unit 7."
canonical: "https://fiveable.me/ap-stats/key-terms/dependent-samples"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 4"
---

# Dependent Samples — AP Stats Definition & Exam Guide

## Definition

Dependent samples are samples whose observations are paired or linked, such as before-and-after measurements on the same subjects or matched pairs, so in AP Stats you analyze the differences with one-sample t procedures instead of two-sample procedures for independent samples.

## What It Is

Dependent samples (often called paired data) are two sets of measurements that are connected observation by observation. The classic setups are before-and-after measurements on the same people, two [treatments](/ap-stats/key-terms/treatments "fv-autolink") applied to the same subject, or a [matched pairs design](/ap-stats/unit-1/experimental-design/study-guide/gsdVWumN3cEYmXOIVv95 "fv-autolink") where subjects are paired by some trait and each member of the pair gets a different treatment. Because each value in one sample has a natural partner in the other, the samples are not independent of each other.

This matters because it changes the procedure you pick. Learning objective 7.6.A asks you to identify the appropriate confidence interval for a [difference of two means](/ap-stats/unit-4/sampling-distributions-for-differences-sample-means/study-guide/hPyIdIhuKKF731eU2qOT "fv-autolink"), and the two-sample t interval in Topic 7.6 (with standard error √(s₁²/n₁ + s₂²/n₂)) only applies when the data come from two independent random samples or a randomized experiment with two groups. With dependent samples, you don't have two separate samples in any meaningful sense. You have one sample of pairs. So you subtract within each pair, treat those differences as a single quantitative variable, and run one-sample t procedures (a t-interval or t-test for the mean difference μd).

## Why It Matters

Dependent samples live in Unit 7 ([Inference for Quantitative Data: Means](/ap-stats/unit-4 "fv-autolink")) and show up most sharply in Topic 7.6, where learning objective 7.6.A makes you choose between procedures. The essential knowledge for 7.6.B says the two-sample conditions require data from two [independent random samples](/ap-stats/key-terms/independent-random-samples "fv-autolink") or a randomized experiment. Paired data fail that condition on purpose, which is exactly why the College Board loves asking about them. The fastest way to lose points on an inference FRQ is to run a two-sample t procedure on paired data, because the independence condition is violated and the standard error formula is wrong. Recognizing dependence is a design question as much as an inference question, so it also connects back to how the study was set up in the first place.

## Connections

### [Independent random samples (Unit 7)](/ap-stats/key-terms/independent-random-samples)

This is the direct opposite of dependent samples, and the whole decision in 7.6.A hinges on telling them apart. Two separately drawn [random samples](/ap-stats/key-terms/random-sample "fv-autolink"), or random assignment to two groups, gives you independent samples and a two-sample t procedure. Paired or matched data give you dependent samples and a one-sample t on the differences.

### Matched pairs design (Unit 3)

Dependent samples are usually born in the experimental design stage. A matched pairs experiment, like pairing students by ability and giving each pair member a different teaching method, deliberately creates linked observations to reduce variability. [Unit 3](/ap-stats/unit-3 "fv-autolink") teaches you to build that design; Unit 7 teaches you to analyze it correctly.

### Confidence Interval for the Difference of Two Means (Unit 7)

The two-sample [interval](/ap-stats/unit-1/representing-quantitative-variable-with-graphs/study-guide/VWtyLVDvjzEgtbAi6v6j "fv-autolink") (x̄₁ − x̄₂) ± t*√(s₁²/n₁ + s₂²/n₂) from Topic 7.6 is the procedure you must NOT use on dependent samples. Knowing this term well is basically knowing when that formula is off-limits.

### [Margin of Error (Unit 7)](/ap-stats/key-terms/margin-of-error)

The standard error inside the margin of error changes with the design. Independent samples combine two variances; dependent samples use the standard deviation of the differences divided by √n. Same t* idea, completely different SE.

## On the AP Exam

Multiple choice questions test whether you can spot the design. A stem describing students matched by ability with one in each pair getting a new teaching method is signaling paired data, and the correct answer involves a one-sample t procedure on the differences. Compare that to a stem with 45 randomly selected vegetarians and 50 non-vegetarians, which is two independent samples and a two-sample t interval. No released FRQ has used the phrase "dependent samples" verbatim, but inference FRQs routinely require you to name the correct procedure and check conditions, and the independence check in 7.6.B is where paired data trips people up. On an FRQ, identifying the wrong procedure usually tanks the whole part, so train yourself to ask one question first. Are these measurements linked one-to-one, or are they two unrelated groups?

## dependent samples vs Independent random samples

Independent samples come from two separate groups with no connection between individual observations, like vegetarians versus non-vegetarians selected separately. Dependent samples have a one-to-one link between observations, like the same person measured twice or matched pairs. The test is simple. If you can sensibly subtract within each row of the data table, the samples are dependent. Independent samples get two-sample t procedures; dependent samples get one-sample t procedures on the differences.

## Key Takeaways

- Dependent samples are paired or linked observations, such as before-and-after measurements on the same subjects or matched pairs in an experiment.
- With dependent samples, you compute the difference within each pair and run one-sample t procedures on those differences, not two-sample procedures.
- The two-sample t interval in Topic 7.6 requires two independent random samples or a randomized experiment, so paired data violate its conditions.
- Common signals of dependence include the same subjects measured twice, twins or matched partners, and two treatments applied to one individual.
- Choosing between paired and two-sample procedures is exactly what learning objective 7.6.A tests, and picking wrong on an FRQ costs you the inference points.

## FAQs

### What are dependent samples in AP Stats?

Dependent samples are two sets of measurements linked observation by observation, like pre-test and post-test scores for the same students or matched pairs in an experiment. You analyze them by taking the difference within each pair and running one-sample t procedures.

### Do you use a two-sample t-test for dependent samples?

No. Two-sample t procedures require two independent random samples or a randomized two-group experiment. For dependent samples you subtract within pairs and use a one-sample t-test or t-interval on the mean difference μd.

### How do I tell if samples are dependent or independent on the AP exam?

Ask whether each observation in one group has a specific partner in the other. Same subjects measured twice, matched pairs, and twins mean dependent. Two separately selected groups, like 45 vegetarians and 50 non-vegetarians chosen independently, mean independent.

### Are matched pairs the same as dependent samples?

Yes, a matched pairs design produces dependent samples. Whether the pairing comes from measuring the same person twice or matching two similar subjects, the resulting data are paired and get one-sample t procedures on the differences.

### What conditions do dependent samples violate for a two-sample interval?

They violate the independence condition in 7.6.B, which requires data from two independent random samples or a randomized experiment. Because paired observations are related, the two-sample standard error √(s₁²/n₁ + s₂²/n₂) doesn't describe the data correctly.

## 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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