---
title: "Sampling Without Replacement — AP Stats Definition"
description: "Sampling without replacement means each item can be picked only once. Learn how it shrinks the standard deviation of p̂ and why the 10% condition exists in AP Stats."
canonical: "https://fiveable.me/ap-stats/key-terms/without-replacement"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 1"
---

# Sampling Without Replacement — AP Stats Definition

## Definition

Sampling without replacement is a method where each item from a population can be selected only once and is not returned for later picks. In AP Stats it makes draws dependent, slightly shrinking the standard deviation of p̂, which is why the 10% condition exists (sample size less than 10% of the population).

## What It Is

[Sampling without replacement](/ap-stats/key-terms/sampling-without-replacement "fv-autolink") means once you pick an item from the population, it's out. You can't pick it again. That's how almost all real sampling works. If you call 150 voters for a poll, you don't call the same voter twice. The CED defines it directly in [Topic 3.3](/ap-stats/unit-3/constructing-confidence-interval-for-population-proportion/study-guide/rYCExLGlPYtMNFiJOWAR "fv-autolink") (DAT-2.C.1): when an item can be selected only once, that's sampling without replacement; when it can be selected more than once, that's sampling with replacement.

Here's the catch that shows up in [Unit 5](/ap-stats/unit-5 "fv-autolink"). The famous formula σp̂ = √(p(1-p)/n) technically assumes independent draws, which means sampling *with* replacement. When you sample without replacement, each draw changes the population left behind, so the draws aren't independent and the true standard deviation of p̂ is slightly **smaller** than the formula says. The fix is the 10% condition. If your sample is less than 10% of the population, the difference is negligible and you can use the formula anyway. That's the whole reason the 10% condition exists.

## Why It Matters

This term lives in two places. In [Unit 3](/ap-stats/unit-3 "fv-autolink") (Topic 3.3), it's a basic vocabulary distinction you need to identify sampling methods under [AP Stats](/ap-stats "fv-autolink") 3.3.A. In Unit 5 (Topic 5.5), it does real mathematical work under AP Stats 5.5.A. The essential knowledge there says it plainly: if sampling is done without replacement, the standard deviation of the sample proportion is smaller than √(p(1-p)/n), but if the sample size is less than 10% of the population size, the difference is negligible. So every time you check the 10% condition before a confidence interval or significance test in Units 6 and 7, you're really answering the question "did we sample without replacement from a big enough population that we can pretend the draws were independent?" Understanding *why* the condition exists, not just memorizing it, is what separates a 5-level answer from a recited checklist.

## Connections

### [Sampling with replacement (Units 3 & 5)](/ap-stats/key-terms/sampling-with-replacement)

This is the flip side of the same coin. With replacement, every draw is independent because the [population](/ap-stats/key-terms/population "fv-autolink") never changes, so σp̂ = √(p(1-p)/n) is exactly right. Without replacement, draws are dependent and that formula slightly overstates the spread.

### Sampling Distributions for Sample Proportions (Unit 5)

The [10% condition](/ap-stats/key-terms/10percent-condition "fv-autolink") you check in Topic 5.5 is a direct consequence of sampling without replacement. As long as n is under 10% of the population, you're allowed to use the with-replacement formula even though you didn't actually replace anyone.

### [Random Number Generator (Unit 3)](/ap-stats/key-terms/random-number-generator)

A standard mechanism for getting an SRS is to number every individual, generate random numbers, and ignore repeats. That "ignore repeats" step is sampling without replacement in action. You skip duplicates precisely so no one gets selected twice.

### [Large Counts Condition (Unit 5)](/ap-stats/key-terms/large-counts-condition)

These two conditions answer different questions, so don't merge them. [Large Counts](/ap-stats/key-terms/large-counts "fv-autolink") (np ≥ 10 and n(1-p) ≥ 10) checks whether the shape of p̂'s distribution is approximately normal. The 10% condition, born from sampling without replacement, checks whether the standard deviation formula is trustworthy.

## On the AP Exam

On multiple choice, this term shows up in two flavors. Unit 3 questions ask you to identify or evaluate a sampling method (AP Stats 3.3.A and 3.3.B), where "each person can be chosen only once" is the giveaway. Unit 5 questions are sneakier. A typical stem gives you a sample and a population, like 150 voters from a town of 1,200, or 200 students from a university of 5,000, and asks what's true about the standard deviation of p̂. You need to check whether n is under 10% of the population and know that without replacement, the true standard deviation is smaller than (or, when the 10% condition holds, approximately equal to) √(p(1-p)/n). On FRQs, the idea appears through probability problems involving random assignment or selection where outcomes can't repeat, like the 2017 FRQ where four people are randomly split into two groups. There, each assignment removes a person from the pool, so probabilities change with each pick. The most common scored move, though, is correctly checking and *interpreting* the 10% condition during inference procedures, which connects back to AP Stats 5.5.C.

## without replacement vs Sampling with replacement

With replacement means an item goes back into the pool and can be picked again, so every draw is independent and the population never changes. Without replacement means each item can be picked only once, so each draw shrinks the remaining pool and the draws are dependent. The exam consequence is in Unit 5. The standard deviation formula σp̂ = √(p(1-p)/n) is built for with-replacement (independent) sampling. Real surveys sample without replacement, which makes the true standard deviation slightly smaller. The 10% condition is the bridge that lets you use the simple formula anyway.

## Key Takeaways

- Sampling without replacement means each item from the population can be selected only once and is not returned for future draws.
- Without replacement, draws are dependent because each selection changes the population that remains.
- When sampling without replacement, the true standard deviation of p̂ is smaller than the value given by √(p(1-p)/n).
- The 10% condition exists because of sampling without replacement. If n is less than 10% of the population, the difference in standard deviation is negligible and the formula is fine.
- Almost all real-world surveys and polls sample without replacement, which is why the 10% condition gets checked in nearly every inference problem about proportions.
- Don't confuse the 10% condition (justifies the standard deviation formula) with the Large Counts condition (justifies approximate normality of p̂).

## FAQs

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

It's a sampling method where each item can be selected only once and is not returned to the population for later selections (CED Topic 3.3, DAT-2.C.1). Calling 150 different voters for a poll is sampling without replacement, since nobody gets surveyed twice.

### Does sampling without replacement make the standard deviation of p̂ bigger or smaller?

Smaller. The CED states that without replacement, the standard deviation of the sample proportion is smaller than √(p(1-p)/n). If the sample is less than 10% of the population, the difference is negligible, so you can still use the formula.

### Is the 10% condition the same as the Large Counts condition?

No, they check different things. The 10% condition (n less than 10% of the population) justifies using the standard deviation formula despite sampling without replacement. The Large Counts condition (np ≥ 10 and n(1-p) ≥ 10) justifies treating the sampling distribution of p̂ as approximately normal.

### How is sampling without replacement different from sampling with replacement?

With replacement, items go back in the pool and can be chosen again, so draws are independent. Without replacement, each item can be chosen only once, so draws are dependent because the pool shrinks with every pick.

### Why do random number generator methods say to ignore repeats?

Because ignoring repeats is how you sample without replacement. If your generator spits out the same individual's number twice, skipping the duplicate guarantees that person is only selected once, which is what an SRS in practice requires.

## Related Study Guides

- [1.11 Random Sampling](/ap-stats/unit-1/random-sampling-data-collection/study-guide/nQz8XwRMmIKKBS59qrew)

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