10% Condition in AP Statistics

In AP Statistics, the 10% condition says that when you sample without replacement, your sample size n should be no more than 10% of the population size N (n ≤ 0.10N). This keeps observations approximately independent so the standard deviation formulas for inference work.

What is the 10% Condition?

The 10% condition is a check you run before doing inference (a confidence interval or significance test) whenever your sample was taken without replacement. You verify that the sample is no more than 10% of the population, or n ≤ 0.10N.

Why bother? Sampling without replacement technically breaks independence. Once you pick one person, they can't be picked again, so each draw slightly changes what's left. If the population is huge compared to the sample, that change is so tiny it doesn't matter. The 10% condition is your proof that the population is "big enough" to treat each observation as independent. Think of it like scooping a cup of water out of a swimming pool. You barely changed the pool, so the next cup looks just like the first.

Why the 10% Condition matters in AP® Statistics

The 10% condition shows up every time you set up inference from a sample taken without replacement. That includes setting up a test for a population proportion (Topic 3.5), building a confidence interval for the difference between two population proportions (Topic 3.10), and setting up a test for a population mean or mean difference (Topic 4.4). In each case it's part of checking the independence condition, alongside randomness and a sample-size check.

It matters because the standard deviation (and standard error) formulas for sampling distributions assume independent observations. If the sample is too large a chunk of the population, those formulas overstate the true variability, and your intervals and p-values stop being trustworthy. On the AP exam, skipping or mis-stating this check costs you points in the "conditions" part of an inference FRQ.

How the 10% Condition connects across the course

Independence Condition (Units 3-4)

The 10% condition isn't a separate requirement floating on its own. It's how you justify independence when you sampled without replacement. Independence is the goal, and the 10% check is the evidence.

Randomness Condition and Simple Random Samples (Units 3-4)

Randomness and the 10% condition do different jobs. A simple random sample lets you generalize to the population. The 10% check makes sure that drawing that SRS without replacement didn't wreck independence. You usually state both, back to back.

Two-Sample Proportion Intervals (Unit 3)

With two samples, you run the 10% check twice, once for each population. For example, 250 households from County 1 and 300 from County 2 means County 1 needs at least 2,500 households and County 2 needs at least 3,000.

One Sample T-test and Dependent Samples (Unit 4)

The same check carries over from proportions to means. For a one-sample t-test, compare n to the population. For matched pairs (dependent samples), compare the number of pairs to the population of pairs, since each difference counts as one observation.

Is the 10% Condition on the AP® Statistics exam?

Multiple choice: Expect conceptual stems like "What does the 10% condition prevent when testing a population mean?" or "What is the primary reason for ensuring the sample size is no more than 10% of the population when sampling without replacement?" The answer always points back to approximate independence of observations. It is not about normality, bias, or the sample being random.

FRQs: The 10% condition was part of 2025 FRQ Q4, and it shows up any time an FRQ asks you to "check the conditions" for a test or interval. To earn credit, you need to:

• State the condition in context (e.g., "250 is less than 10% of all households in County 1").
• Write it as an inequality or with an actual number. For example, write "the county has at least 2,500 households" rather than just "the population is large."
• Do it for each sample in a two-sample problem.

A bare "10% condition is met" with no context usually won't earn full credit.

The 10% Condition vs Large Counts Condition

Both involve the number 10, which is why they get mixed up. The 10% condition compares sample size to population size (n ≤ 0.10N) and protects independence. The Large Counts condition for proportions checks that expected successes and failures are at least 10 (np ≥ 10 and n(1 − p) ≥ 10) and protects approximate normality of the sampling distribution. One is a percent of the population, and the other is a count of successes and failures.

Key things to remember about the 10% Condition

• The 10% condition says that when sampling without replacement, the sample size should be no more than 10% of the population size.

• Its purpose is to make observations approximately independent, which the standard deviation formulas for sampling distributions require.

• On a two-sample problem, you must check the 10% condition separately for each population.

• Always state the condition in context with real numbers, such as saying a county has at least 2,500 households when n = 250.

• The 10% condition supports independence, while the Large Counts condition supports normality, so don't swap them.

• If you sampled with replacement or randomly assigned treatments in an experiment, the 10% condition isn't the check you need.

Frequently asked questions about the 10% Condition

What is the 10% condition in AP Stats?

It's the check that a sample taken without replacement is no more than 10% of the population (n ≤ 0.10N). Meeting it lets you treat observations as approximately independent, which inference formulas assume.

Does the 10% condition check for normality?

No. The 10% condition is about independence. Normality is handled by the Large Counts condition for proportions (at least 10 successes and failures) or by sample size and data shape for means.

How is the 10% condition different from the Large Counts condition?

The 10% condition compares sample size to population size and protects independence. Large Counts checks that np and n(1 − p) are each at least 10 and protects the approximate normality of the sampling distribution of p̂.

Do you need the 10% condition for an experiment?

Usually not. In an experiment with random assignment, independence comes from the random assignment itself, not from sampling without replacement. So the 10% check applies when you sample from a population.

How do you write the 10% condition on an FRQ?

Put it in context with numbers. For example, write "It is reasonable to assume there are at least 3,000 households in County 2, so 300 is less than 10% of the population." For two samples, write one statement for each population.