Sampling without replacement is selecting individuals from a population so that each one can be chosen only once, which makes selections slightly dependent. On the AP Stats exam, you handle this by checking the 10% condition (n ≤ 10% of N) before doing inference like a one-sample z-interval for a proportion.
Sampling without replacement means once you pick someone from the population, they're out of the pool. You can't pick them again. This is how almost all real sampling works (you wouldn't survey the same person twice), but it creates a small statistical wrinkle. Each draw changes the makeup of the remaining population, so the draws aren't truly independent. Pull one red marble from a jar and the probability of red on the next draw shifts.
Here's the AP-relevant part. Most inference formulas, like the standard error of p̂, assume independent observations. Sampling without replacement technically breaks that assumption. The fix is the 10% condition. If your sample size n is no more than 10% of the population size N, the dependence is so tiny that you can treat the observations as independent anyway. That's why every confidence interval and significance test for proportions asks you to verify n ≤ 10% of N when sampling without replacement (AP Stats 6.2.B).
This term lives in Topic 6.2 (Constructing a Confidence Interval for a Population Proportion) and directly supports learning objective AP Stats 6.2.B, verifying the conditions for a one-sample z-interval for a proportion. The CED's essential knowledge spells it out. To check independence, data should come from a random sample or randomized experiment, and when sampling without replacement, you check that n ≤ 10% of N. If you skip or botch this condition check on an FRQ, you lose points even if your interval calculation is perfect. The same idea echoes through every inference procedure in Units 6 through 9, so understanding why the 10% condition exists (not just memorizing it) pays off across the whole second half of the course.
Independence (Units 4-6)
Sampling without replacement is the reason independence is in question at all. Without replacement, each pick slightly changes the odds for the next pick. The 10% condition is your permission slip to treat draws as independent anyway when the sample is small relative to the population.
Sampling Distribution (Unit 5)
The formula for the standard deviation of p̂ assumes independent observations. When you sample without replacement from a population at least 10 times your sample size, the sampling distribution behaves almost exactly as if you'd sampled with replacement, so the Unit 5 formulas still apply.
Confidence Interval (Unit 6)
Before you build a one-sample z-interval for a proportion, you verify conditions. The 10% check exists specifically because real surveys sample without replacement. It's the second box you check, right after random sampling.
Population (Unit 3)
The 10% condition compares your sample size n to the population size N. That means you need to clearly identify the population in context, a Unit 3 skill that follows you into every inference problem.
You'll almost never see a question asking you to define sampling without replacement directly. Instead, it shows up inside the conditions check. On inference FRQs, the standard rubric expects you to name the procedure, then verify conditions, and the independence check is where this term does its work. You write something like "since we are sampling without replacement, we check that n ≤ 10% of N. Since 50 ≤ 10% of all 2,000 students at the school, independence is reasonable." Multiple-choice questions test it by giving a scenario where the sample is too large relative to the population (say, surveying 60 people from a club of 80) and asking which condition is violated. The key move is always the same. Spot that sampling happened without replacement, then show the 10% comparison with actual numbers in context.
With replacement means you put each selected individual back, so they could be chosen again and every draw has identical, truly independent probabilities. Without replacement means no repeats, so probabilities shift slightly after each draw. AP Stats inference assumes the math of with-replacement sampling, and the 10% condition is the bridge that lets without-replacement data use it. Real-world surveys are almost always without replacement, which is exactly why the condition check exists.
Sampling without replacement means each individual can be selected only once, which makes the selections slightly dependent on each other.
The 10% condition (n ≤ 10% of N) lets you treat observations as approximately independent when sampling without replacement.
Checking the 10% condition is part of verifying independence under learning objective AP Stats 6.2.B before constructing a confidence interval for a proportion.
On FRQs, show the actual comparison with numbers, such as "100 ≤ 10% of 1,500," rather than just naming the condition.
The same 10% check applies to inference for proportions, means, and slopes throughout Units 6-9, so master it once and reuse it everywhere.
It's selecting individuals from a population where each one can only be chosen once and isn't returned to the pool. It matters in Unit 6 because it makes observations slightly dependent, which is why you check the 10% condition before doing inference.
Technically yes, but practically it usually doesn't matter. If your sample is no more than 10% of the population (n ≤ 10% of N), the dependence is so small that the CED lets you treat observations as independent for inference.
With replacement, selected individuals go back into the pool and can be chosen again, so every draw is truly independent. Without replacement, there are no repeats, so each draw slightly changes the probabilities for the next one. Real surveys are almost always without replacement.
The standard error formula for p̂, which is √(p̂(1-p̂)/n), assumes independent observations. When sampling without replacement, that assumption is only approximately true, and the 10% condition guarantees the approximation is good enough to use the formula.
No. The 10% condition applies when you take a random sample without replacement from a population. In a randomized experiment, random assignment to treatments is what supports the inference, so the n ≤ 10% of N check isn't required.