---
title: "10% Rule — AP Stats Definition & When to Check It"
description: "The 10% Rule says your sample should be no more than 10% of the population when sampling without replacement. It's the independence check for AP Stats inference."
canonical: "https://fiveable.me/ap-stats/key-terms/10percent-rule"
type: "key-term"
subject: "AP Statistics"
unit: "Unit XXGRYZTH6sTfEcdI"
---

# 10% Rule — AP Stats Definition & When to Check It

## Definition

The 10% Rule (or 10% condition) is an AP Statistics check for independence: when sampling without replacement, the sample size n must be no more than 10% of the population N (n ≤ 0.10N) so observations can be treated as approximately independent for inference procedures like the chi-square goodness-of-fit test.

## What It Is

The 10% Rule is the condition you check to justify treating observations as independent when you [sample](/ap-stats/unit-1/random-sampling-data-collection/study-guide/nQz8XwRMmIKKBS59qrew "fv-autolink") **[without replacement](/ap-stats/key-terms/without-replacement "fv-autolink")**. Here's the problem it solves. Technically, every time you pull someone out of a population without putting them back, the probabilities for everyone left shift slightly. That violates independence, which the formulas for standard errors and chi-square statistics assume. The fix is practical, not magical. If your sample is small relative to the population (n ≤ 10% of N), those probability shifts are so tiny that pretending the observations are independent barely changes anything.

In Topic 8.2, the CED makes this an explicit step in verifying conditions for a chi-square goodness-of-fit test ([AP Stats](/ap-stats "fv-autolink") 8.2.E). The independence check has two parts. First, the data must come from a random sample or randomized experiment. Second, when sampling without replacement, you confirm n ≤ 10% N. So if you survey 50 students from a school of 1,200, you check 50 ≤ 120. Condition met, and you say so in writing.

## Why It Matters

The 10% Rule lives in **Unit 8: Inference for [Categorical Data](/ap-stats/key-terms/categorical-data "fv-autolink"): Chi-Square**, specifically Topic 8.2, where learning objective **AP Stats 8.2.E** requires you to verify conditions before running a [chi-square goodness-of-fit test](/ap-stats/unit-3/setting-up-chi-square-test-for-homogeneity-or-independence/study-guide/tZimSvE0pNy9ak5fjCmh "fv-autolink"). But this is one of those conditions that follows you through the whole second half of the course. Any inference procedure built on sampling without replacement (confidence intervals and tests for proportions, means, and chi-square) leans on the same logic. On the exam, condition-checking is scored, not assumed. Skipping the 10% check, or checking it without showing the numbers, costs points on inference FRQs. It's also a favorite MCQ angle, since questions love asking which condition a given scenario violates.

## Connections

### Independence Assumption (Unit 8)

The 10% Rule isn't a separate idea, it's HOW you verify [independence](/ap-stats/key-terms/independence "fv-autolink") when sampling without replacement. Random selection plus n ≤ 10% N together justify treating observations as independent in the CED's condition checklist.

### Sampling Distribution (Unit 5)

The formulas for standard deviation of a sampling distribution assume independent observations. The 10% Rule is what lets you use those formulas even though real-world sampling almost never replaces people back into the [population](/ap-stats/key-terms/population "fv-autolink").

### [Expected Count (Unit 8)](/ap-stats/key-terms/expected-count)

The 10% Rule and the [large counts condition](/ap-stats/key-terms/large-counts-condition "fv-autolink") (all expected counts greater than 5) are the two checks you run for a chi-square goodness-of-fit test. One handles independence, the other handles the shape of the chi-square distribution. You need both.

### [Random Sample (Unit 3)](/ap-stats/key-terms/random-sample)

The 10% Rule is the second half of the independence check; random sampling is the first half. A sample can pass the 10% check and still be useless if it wasn't randomly selected, so always state both.

## On the AP Exam

On the AP Stats exam, the 10% Rule shows up wherever conditions get checked. In multiple choice, expect stems that give you a sample size and a population size and ask whether inference conditions are met, or that describe a scenario violating one condition and ask you to spot which. In free response, any full significance test or confidence interval question expects a written condition check, and graders want to see the actual comparison (for example, "n = 40 ≤ 10% of N = 500"), not just the phrase "conditions are met." Watch for questions that mix up the conditions on purpose. A practice question might give you expected counts of 3 and 4 in some categories; that violates the large counts condition, not the 10% Rule, and you have to know the difference to answer correctly.

## 10% Rule vs Large Counts Condition

Both are conditions for chi-square inference, but they protect different things. The 10% Rule (n ≤ 10% N) protects **independence** when sampling without replacement. The large counts condition (all expected counts greater than 5) protects the **shape** approximation, making sure the chi-square distribution actually fits your test statistic. A common exam trap gives you small expected counts and tempts you to blame the 10% Rule. Check them separately, and name which one fails.

## Key Takeaways

- The 10% Rule states that when sampling without replacement, the sample size should be no more than 10% of the population (n ≤ 0.10N).
- Its purpose is to justify treating observations as approximately independent, since sampling without replacement technically violates independence.
- Per AP Stats 8.2.E, the full independence check is random sampling (or random assignment) plus the 10% condition.
- The 10% Rule is separate from the large counts condition; 10% handles independence while large counts (expected counts greater than 5) handles the shape of the chi-square distribution.
- On FRQs, show the actual numbers when checking the condition (write n ≤ 10% of N with values plugged in) instead of just asserting that conditions are met.
- The 10% Rule applies across inference procedures, not just chi-square, so the same check appears for proportions and means whenever sampling is done without replacement.

## FAQs

### What is the 10% Rule in AP Stats?

It's the condition that your sample size n should be no more than 10% of the population size N when sampling without replacement. It lets you treat observations as approximately independent for inference, including the chi-square goodness-of-fit test in Topic 8.2.

### Does the 10% Rule apply when sampling with replacement?

No. With replacement, each draw doesn't change the population, so observations are already independent and there's nothing to fix. The 10% Rule only matters for sampling without replacement, which is how almost all real surveys work.

### Is the 10% Rule the same as the large counts condition?

No, and the exam loves testing this. The 10% Rule (n ≤ 10% N) checks independence, while the large counts condition (all expected counts greater than 5) checks that the chi-square shape approximation is valid. A sample of 50 from a population of 5,000 passes the 10% check but could still fail large counts if a category's expected count is only 3.

### Why does the sample need to be less than 10% of the population?

Removing people without replacement changes the probabilities for everyone left. If the sample is a small slice of the population, those changes are negligible, so independence-based formulas still work well. 10% is the agreed cutoff where the approximation stays accurate.

### Do I have to write out the 10% condition on an FRQ?

Yes, if the data came from sampling without replacement. State the comparison with actual numbers, like "n = 60 ≤ 10% of N = 1,000," alongside the random sampling check. Just writing "conditions met" without the verification doesn't earn the condition-checking component.

## Related Study Guides

- [Legacy AP Statistics Topic: Chi-Square Goodness of Fit Setup](/ap-stats/unit-XXGRYZTH6sTfEcdI/setting-up-chi-square-goodness-fit-test/study-guide/2W5HT2MbSAG4Ty1buvCd)

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