---
title: "Ten Percent Condition — AP Stats Definition & Exam Guide"
description: "The ten percent condition requires n ≤ 10% of the population so sampling without replacement still acts independent. Essential for checking conditions in AP Stats."
canonical: "https://fiveable.me/ap-stats/key-terms/ten-percent-condition"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 3"
---

# Ten Percent Condition — AP Stats Definition & Exam Guide

## Definition

The ten percent condition is the AP Statistics check that the sample size n is less than 10% of the population size, which makes sampling without replacement close enough to independent that the standard deviation formula σp̂ = √(p(1-p)/n) is still valid.

## What It Is

The ten percent condition says your [sample](/ap-stats/unit-1/random-sampling-data-collection/study-guide/nQz8XwRMmIKKBS59qrew "fv-autolink") should be less than 10% of the [population](/ap-stats/key-terms/population "fv-autolink") you're sampling from. Why? The standard deviation formula for a sample proportion, σp̂ = √(p(1-p)/n), assumes independent observations, which technically only happens when you sample **with** replacement. Real surveys almost never do that. Once you pull someone out of the population without putting them back, the makeup of the remaining population shifts slightly, so the trials aren't truly independent.

Here's the intuition. If you scoop 5 candies from a bag of 10,000, the bag barely notices, so each pick is basically independent. If you scoop 5 from a bag of 8, every pick dramatically changes what's left. The CED's essential knowledge for [AP Stats](/ap-stats "fv-autolink") 5.5.A spells this out. Sampling without replacement actually makes the true standard deviation a bit *smaller* than the formula gives, but if n is less than 10% of the population, that difference is negligible and you can use the formula with a clear conscience.

## Why It Matters

The ten percent condition lives in Topic 5.5 (Sampling Distributions for Sample Proportions) and directly supports AP Stats 5.5.A, determining the [parameters](/ap-stats/key-terms/parameter "fv-autolink") of the sampling distribution of p̂. You can't claim σp̂ = √(p(1-p)/n) unless this condition holds, because that formula assumes independence. It also feeds into 5.5.C, since interpreting probabilities from a sampling distribution only makes sense if the distribution's parameters were legitimately calculated in the first place. Beyond [Unit 5](/ap-stats/unit-5 "fv-autolink"), this condition becomes one of the standard checks you write out every time you build a confidence interval or run a significance test in Units 6 and 7. Skipping it on an FRQ costs you points even if your math is perfect.

## Connections

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

These two conditions answer different questions about the same sampling distribution. The ten percent condition justifies the standard deviation formula ([independence](/ap-stats/key-terms/independence "fv-autolink")), while the Large Counts Condition (np ≥ 10 and n(1-p) ≥ 10) justifies calling the shape approximately normal. You need both before using normal calculations with p̂, and the exam loves checking whether you know which condition does which job.

### [Without replacement (Units 4-5)](/ap-stats/key-terms/without-replacement)

The ten percent condition only exists because real sampling is done [without replacement](/ap-stats/key-terms/without-replacement "fv-autolink"). If you sampled with replacement, observations would be truly independent and you'd never need this check. The condition is essentially a permission slip to treat without-replacement sampling as if it were with-replacement.

### [Expected successes and expected failures (Unit 5)](/ap-stats/key-terms/expected-successes-and-expected-failures)

Expected successes (np) and expected failures (n(1-p)) are what you actually compute for the [Large Counts Condition](/ap-stats/key-terms/large-counts-condition "fv-autolink"), the ten percent condition's partner check. Together they form the full conditions checklist for the sampling distribution of p̂, a checklist that carries straight into inference for proportions in Unit 6.

### Mean of the sampling distribution (Unit 5)

Here's a detail worth knowing. Violating the ten percent condition messes up the standard deviation of p̂, but the mean μp̂ = p stays correct either way. The condition is purely about getting the spread right, not the center.

## On the AP Exam

On multiple choice, expect questions that give you a sample size and a population size and ask whether the standard deviation formula for p̂ is appropriate, or ask what condition justifies independence when sampling without replacement. On free response, the ten percent condition is part of the conditions check you write before any probability calculation with a sampling distribution or any inference procedure for proportions. The move that earns credit is explicit. Write something like "n = 50 is less than 10% of the population of 2,000 students, so observations can be treated as independent." A common point-loser is checking the Large Counts Condition but forgetting the ten percent check entirely, or stating the condition without comparing actual numbers. No released FRQ uses the phrase "ten percent condition" verbatim, but conditions checks like this one are scored on nearly every inference FRQ.

## ten percent condition vs Large Counts Condition

Both are conditions you check before working with the sampling distribution of p̂, so it's easy to blur them together. The ten percent condition (n < 10% of the population) is about independence, and it justifies using σp̂ = √(p(1-p)/n) when sampling without replacement. The Large Counts Condition (np ≥ 10 and n(1-p) ≥ 10) is about shape, and it justifies treating the sampling distribution as approximately normal. One checks the sample against the population size; the other checks expected successes and failures. They are not interchangeable, and an FRQ response needs both, each tied to its own purpose.

## Key Takeaways

- The ten percent condition requires the sample size to be less than 10% of the population size, which lets you treat observations as independent even when sampling without replacement.
- It exists because the standard deviation formula σp̂ = √(p(1-p)/n) assumes independence, which technically requires sampling with replacement.
- Sampling without replacement makes the true standard deviation slightly smaller than the formula says, but the difference is negligible when n is under 10% of the population.
- The ten percent condition checks independence and justifies the standard deviation formula, while the Large Counts Condition checks shape and justifies approximate normality. They are separate checks with separate jobs.
- Violating the condition affects the standard deviation of p̂, not its mean, since μp̂ = p holds regardless.
- On FRQs, show the actual comparison (for example, "100 < 10% of 5,000") rather than just naming the condition, because explicit checks are what earn condition points.

## FAQs

### What is the ten percent condition in AP Stats?

It's the requirement that your sample size be less than 10% of the population size. When it holds, sampling without replacement is close enough to independent that you can use the standard deviation formula σp̂ = √(p(1-p)/n) for the sampling distribution of a sample proportion.

### Is the ten percent condition the same as the Large Counts Condition?

No. The ten percent condition checks independence by comparing sample size to population size, while the Large Counts Condition checks normality by requiring np ≥ 10 and n(1-p) ≥ 10. You typically need to verify both, and each justifies a different claim.

### Does violating the ten percent condition make your results completely wrong?

Not completely. The mean μp̂ = p is still correct, but the formula overstates the standard deviation. The true spread of p̂ is smaller than √(p(1-p)/n) gives you, so probability calculations based on the formula become inaccurate.

### Why is it 10% and not some other number?

It's a practical cutoff, not a magic threshold. Below 10%, the gap between the formula's standard deviation and the true (smaller) without-replacement standard deviation is small enough to ignore. The CED's essential knowledge for AP Stats 5.5.A describes the difference as negligible at that point.

### Do I need the ten percent condition if sampling is done with replacement?

No. Sampling with replacement gives you truly independent observations, so the formula σp̂ = √(p(1-p)/n) holds exactly without any check. The ten percent condition only matters for sampling without replacement, which is what almost all real surveys use.

## Related Study Guides

- [3.2 Sampling Distributions for Sample Proportions](/ap-stats/unit-3/sampling-distributions-for-sample-proportions/study-guide/Ezxev8MPpv3mFKjV4Gq3)

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