---
title: "Sampling Without Replacement — AP Stats Definition"
description: "Sampling without replacement means each item can be selected only once, changing probabilities as you go. It's why the 10% condition exists in AP Stats inference."
canonical: "https://fiveable.me/ap-stats/key-terms/sampling-without-replacement"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 3"
---

# Sampling Without Replacement — AP Stats Definition

## Definition

Sampling without replacement is a sampling method in which each item from a population can be selected only once (DAT-2.C.1), so each draw changes the probabilities for the next draw. On the AP exam, it's the reason you check the 10% condition (n ≤ 10% of N) before doing inference.

## What It Is

Sampling without replacement means once you pick an individual from the [population](/ap-stats/key-terms/population "fv-autolink"), you don't put it back. It can't be chosen again. The CED defines this directly in [Topic 3.3](/ap-stats/unit-3/random-sampling-data-collection/study-guide/nQz8XwRMmIKKBS59qrew "fv-autolink") (DAT-2.C.1), alongside its opposite, sampling with replacement, where the same item can show up multiple times.

Here's why it matters mathematically. When you don't replace, each selection shrinks the pool, so the probability of picking any remaining individual changes after every draw. Pull 5 patients from a population of 20 and the second pick has 19 options, not 20. The draws are no longer independent of each other. That's a problem, because most [AP Stats](/ap-stats "fv-autolink") inference formulas assume independence. The fix is simple. If your sample is small relative to the population (no more than 10% of it), the probabilities barely change between draws, and you can treat the observations as approximately independent. That's the famous 10% condition, and sampling without replacement is the entire reason it exists.

## Why It Matters

This term lives in Topic 3.3 (Random Sampling and Data Collection) under LO 3.3.A, where you have to identify sampling methods from a study description, and LO 3.3.B, where you explain why a method is or isn't appropriate. But it doesn't stay in [Unit 3](/ap-stats/unit-3 "fv-autolink"). It quietly follows you through every inference unit. In Topic 8.5, LO 8.5.C explicitly requires that 'when sampling without replacement, check that n ≤ 10% N' before running a chi-square test for homogeneity or [independence](/ap-stats/key-terms/independence "fv-autolink"). The same independence check appears in the conditions for proportion and mean inference too. So this one Unit 3 definition is the backbone of a condition you'll write out on basically every inference FRQ.

## Connections

### [Sampling with replacement (Unit 3)](/ap-stats/key-terms/sampling-with-replacement)

The direct opposite. With replacement, an item goes back in the pool and can be picked again, so every draw has identical probabilities and the draws are truly independent. [Without replacement](/ap-stats/key-terms/without-replacement "fv-autolink"), the pool shrinks and probabilities shift. When the sample is a small fraction of the population, the two methods give nearly identical results, which is exactly the logic behind the 10% condition.

### [Simple random sample (Unit 3)](/ap-stats/key-terms/simple-random-sample)

An SRS is usually done without replacement. DAT-2.C.2 even notes that one way to build an SRS is using a [random number generator](/ap-stats/key-terms/random-number-generator "fv-autolink") and ignoring repeats, which is sampling without replacement in action. If a study description says repeats are skipped, that's your clue.

### Conditions for chi-square tests (Unit 8)

LO 8.5.C makes you verify random sampling, [large counts](/ap-stats/key-terms/large-counts "fv-autolink"), and the 10% condition (n ≤ 10% N) when sampling without replacement. The 10% check isn't busywork. It's your justification for treating non-independent draws as independent enough for the chi-square math to work.

### Sampling distribution (Unit 5)

The standard deviation formulas for sampling distributions assume independent observations. Sampling without replacement technically breaks that, and it actually makes the true variance slightly smaller than the formula says. As long as n ≤ 10% of N, the formula is close enough to use.

## On the AP Exam

Multiple-choice questions hit this term two ways. First, definitionally, asking you to recognize whether a described method is with or without replacement, like an inspector pulling 3 items from a line of 100 and not putting them back. Second, conceptually, asking what happens to independence or variance when you sample without replacement, and why a small sampling fraction (say, a few hundred households out of 5,000) makes the with/without distinction practically irrelevant. Watch for bootstrap resampling questions too. Bootstrapping samples WITH replacement, and the exam loves testing whether you know the difference.

On FRQs, this term shows up indirectly but constantly. Any time you do inference on data sampled without replacement, you must write the 10% condition check (n ≤ 10% N) as part of verifying conditions. Skipping it costs you points on the 'conditions' component of inference questions, including chi-square setups under LO 8.5.C.

## Sampling without replacement vs Sampling with replacement

Without replacement means each individual can be chosen only once, so the pool shrinks and probabilities change with every draw. With replacement means an individual goes back in and can be chosen again, so every draw is identical and independent. The trap on the AP exam is forgetting that bootstrap resampling uses WITH replacement, while a typical SRS in a study uses WITHOUT replacement and therefore needs the 10% condition for inference.

## Key Takeaways

- Sampling without replacement means each individual can be selected from the population only once, while sampling with replacement allows repeats (DAT-2.C.1).
- When you sample without replacement, each draw changes the probabilities for the next draw, so the observations are not truly independent.
- The 10% condition (n ≤ 10% of N) exists because if the sample is small relative to the population, sampling without replacement behaves almost exactly like sampling with replacement.
- You must check the 10% condition whenever you do inference on data sampled without replacement, including chi-square tests for homogeneity and independence (LO 8.5.C).
- Bootstrap resampling is the big exception you should memorize, because it deliberately samples with replacement to simulate a sampling distribution.
- Sampling without replacement gives a slightly smaller variance for the sample statistic than sampling with replacement, since you can't keep drawing the same extreme value.

## FAQs

### What is sampling without replacement in AP Stats?

It's a sampling method where each individual from the population can be selected only once. Once chosen, an item is not returned to the pool, so the probabilities for later selections change with each draw (Topic 3.3, DAT-2.C.1).

### Does sampling without replacement violate independence?

Technically yes, since each draw changes the probabilities for the next one. But if your sample is no more than 10% of the population, the change is so small you can treat the observations as approximately independent. That's exactly what the 10% condition checks.

### What's the difference between sampling with and without replacement?

With replacement, an item goes back into the population and can be picked again, so every draw is independent with identical probabilities. Without replacement, items stay out once picked, so the pool shrinks. For a sample of 3 from 100 items, the variance of the sample proportion is actually slightly larger with replacement.

### Is a simple random sample done with or without replacement?

Almost always without replacement. The CED even describes building an SRS by generating random numbers and ignoring repeats. That's why the 10% condition shows up in the conditions for nearly every inference procedure using an SRS.

### Why do I check the 10% condition for a chi-square test?

Because chi-square tests assume independent observations, and sampling without replacement weakens independence. LO 8.5.C requires verifying n ≤ 10% N so that the dependence between draws is negligible and the chi-square approximation stays valid.

## Related Study Guides

- [3.3 Random Sampling and Data Collection](/ap-stats/unit-3/random-sampling-data-collection/study-guide/nQz8XwRMmIKKBS59qrew)
- [Unit 3 Overview: Collecting Data](/ap-stats/unit-3/review/study-guide/qzicHbvrrBrvuJWSB5xP)

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