10% Condition

The 10% Condition is an AP Statistics check used when sampling without replacement. It says the sample size n must be at most 10% of the population size N (n ≤ 0.10N) so that observations can be treated as approximately independent, which justifies the standard error formulas used in inference.

What is the 10% Condition?

The 10% Condition is the rule you check whenever data come from sampling without replacement. The requirement is simple, n must be no more than 10% of the population (n ≤ 0.10N). Here's why it exists. The standard error formulas you use for inference assume each observation is independent, like drawing with replacement. But real samples almost never put people back in the pool. Once you remove someone, the population shifts slightly for the next draw, so draws aren't truly independent.

The fix is a size limit. If your sample is a small slice of the population, removing a few individuals barely changes the probabilities for the remaining draws. Sampling without replacement then behaves almost exactly like sampling with replacement, and the independence assumption holds well enough to trust your calculations. Think of it like scooping a cup of water from a swimming pool. The pool doesn't notice. Scoop a cup from a glass, and it definitely does. The 10% Condition guarantees you're working with the pool, not the glass.

Why the 10% Condition matters in AP Statistics

The 10% Condition is one of the most repeated checks in the entire course because it shows up in nearly every inference procedure. It's part of verifying conditions for a one-sample z-test for a proportion (AP Stats 6.4.C), a two-sample t-test for a difference of means (AP Stats 7.8.C, where you check it twice, n₁ ≤ 10%N₁ and n₂ ≤ 10%N₂), and a t-test for the slope of a regression model (AP Stats 9.4.C). It also matters back in Unit 4, because treating a count of successes as binomial (AP Stats 4.11.A) requires independent trials, and the 10% Condition is what rescues independence when you sample without replacement. On the exam, 'verify the conditions' is a standard FRQ task, and forgetting this check costs points on otherwise correct inference work.

How the 10% Condition connects across the course

Sampling Distribution (Units 6-7)

Every inference test rests on a sampling distribution of p̂ or x̄, and the formulas describing that distribution assume independent observations. The 10% Condition is the piece of the puzzle that makes independence believable when you sample without replacement.

Standard Error (Units 6-9)

Standard error formulas like √(p(1-p)/n) are derived assuming independence. If your sample is more than 10% of the population, the true variability is smaller than the formula says, so your SE (and everything built on it) is off.

Binomial Random Variable (Unit 4)

Binomial settings require independent trials with a constant probability of success. Sampling without replacement technically breaks both, but when n ≤ 10% of N, the binomial model is still a good approximation. This is the 10% Condition's first appearance in the course.

1-Prop Z-Test (Unit 6)

This is the classic place you write the 10% check. Under AP Stats 6.4.C, the independence condition has two parts, random sampling and n ≤ 10%N, and you state both before computing the test statistic.

Is the 10% Condition on the AP Statistics exam?

The 10% Condition appears anywhere the exam asks you to verify or identify conditions for inference, which is one of the most predictable scoring points on FRQs. The 2025 exam (FRQ Q4) required this kind of condition check, and multiple-choice questions regularly ask which condition ensures independence in a proportion test or which conditions must be verified before a t-test for slope. To earn credit, do three things. First, name the condition and show the actual comparison, like '50 ≤ 10% of 2,000.' Second, attach it to independence specifically, not normality. Third, for two-sample procedures, check it separately for both samples. One common trap is checking it when it isn't needed. If subjects come from a randomized experiment rather than a random sample from a population, you generally don't apply the 10% Condition because you aren't sampling from a larger population at all.

The 10% Condition vs Large Counts Condition

Both involve the number 10, which is exactly why they get mixed up. The 10% Condition (n ≤ 0.10N) checks INDEPENDENCE when sampling without replacement. The Large Counts Condition (np₀ ≥ 10 and n(1-p₀) ≥ 10) checks that the sampling distribution of p̂ is approximately NORMAL. They answer different questions. One says your sample is a small enough slice of the population; the other says your sample is big enough to produce a normal-shaped sampling distribution. A complete condition check for a proportion test needs both, plus randomness.

Key things to remember about the 10% Condition

• The 10% Condition requires the sample size to be at most 10% of the population size, written as n ≤ 0.10N.

• You only check it when sampling without replacement, because that's the situation where draws stop being independent.

• It justifies independence, not normality, so don't confuse it with the Large Counts Condition (np ≥ 10 and n(1-p) ≥ 10).

• For two-sample procedures like the two-sample t-test, you must check it separately for each sample: n₁ ≤ 10%N₁ and n₂ ≤ 10%N₂.

• It first appears in Unit 4, where it lets you use the binomial model even though sampling without replacement technically violates independent trials.

• On FRQs, show the actual numbers in your check (like '100 ≤ 10% of 5,000') rather than just naming the condition.

Frequently asked questions about the 10% Condition

What is the 10% Condition in AP Stats?

It's the requirement that your sample size be no more than 10% of the population (n ≤ 0.10N) when sampling without replacement. Meeting it means you can treat observations as approximately independent, which the standard error formulas for inference assume.

Is the 10% Condition the same as the Large Counts Condition?

No. The 10% Condition (n ≤ 0.10N) checks independence, while the Large Counts Condition (np₀ ≥ 10 and n(1-p₀) ≥ 10) checks that the sampling distribution is approximately normal. A proportion test needs both, and the exam expects you to keep them straight.

Do I always need to check the 10% Condition?

No. You check it only when sampling without replacement from a population. In a randomized experiment where you assign existing subjects to treatments, you're not drawing from a larger population, so the 10% check generally doesn't apply.

Why does the 10% Condition make sampling without replacement okay?

When the sample is a small fraction of the population, removing individuals barely changes the probabilities for the next draws. So sampling without replacement behaves almost exactly like sampling with replacement, and independence is approximately satisfied.

Where does the 10% Condition show up across the AP Stats course?

It appears in Unit 4 for binomial distributions, Unit 6 for proportion tests (AP Stats 6.4.C), Unit 7 for two-sample mean tests (AP Stats 7.8.C), and Unit 9 for the t-test for slope (AP Stats 9.4.C). Any time you verify inference conditions with a sample drawn without replacement, it's part of the independence check.