The 10% rule in statistics says that when you’re sampling from a population without replacement, your sample size should be no more than 10% of the total population. If that condition is met, you can treat each observation in your sample as roughly independent of the others, even though technically it isn’t. This rule of thumb shows up constantly in AP Statistics courses and is one of the standard conditions you need to check before running many common statistical tests.
Why Independence Matters
Most of the foundational tools in statistics, like the binomial distribution and normal-based confidence intervals, assume that each observation in your sample doesn’t affect the next one. When you flip a coin, the result of one flip has zero influence on the next. That’s true independence.
Sampling from a real population doesn’t work that way. Imagine a jar with 100 marbles, 60 red and 40 blue. If you pull out a red marble and don’t put it back, the jar now has 59 red and 40 blue. The probability of drawing red on your next pull has shifted slightly. Each draw changes the composition of what’s left, which means your draws are technically dependent on each other. This is the core problem the 10% rule addresses.
How the Rule Works in Practice
The fix is simple: if you’re only sampling a small fraction of the population, the composition barely changes between draws, so the dependency is negligible. The 10% rule sets the threshold. As long as your sample is no larger than 10% of the population, you can safely pretend the draws are independent and use standard formulas like the binomial probability model.
Here’s a concrete example. Say you’re surveying people at a mall and you want to ask 100 shoppers a yes-or-no question. For the 10% rule to hold, you’d want at least 1,000 people in the mall that day. That way, each person you survey barely changes the makeup of the remaining crowd, and your 100 responses behave almost exactly like 100 independent coin flips (each with the same probability of “yes”).
In an AP Statistics class, you’ll typically state this condition explicitly: “The sample of 100 is less than 10% of the population of 1,000+ shoppers, so we can assume approximate independence.” Skipping this check is a common way to lose points on free-response questions.
The Math Behind the Threshold
The 10% rule isn’t arbitrary. It connects to something called the finite population correction factor. When you sample from a finite population without replacement, the true standard error of your estimate is slightly smaller than the formula for infinite populations would suggest. The correction factor is the square root of (N − n) / (N − 1), where N is the population size and n is the sample size.
When your sample is small relative to the population, this correction factor is very close to 1, meaning you can ignore it. When your sample crosses roughly 5 to 10% of the population, the correction starts to matter. At a 10% sampling fraction, your standard errors calculated with the simpler infinite-population formula will be about 5% too large. That’s a small enough overestimate that most statisticians consider it acceptable, and since it makes your results slightly conservative rather than too optimistic, it’s a safe direction to be wrong in.
Some statisticians prefer a stricter cutoff. One common alternative is 5% of the population, and some practitioners use 1% as their personal threshold. The tighter the cutoff, the closer your approximation gets to reality. But for introductory courses and most practical work, 10% is the standard benchmark.
Where You’ll Use It
The 10% rule comes up in three main situations:
- Binomial problems: When you want to model the number of “successes” in a sample as a binomial random variable, you need independent trials. The 10% rule lets you treat sampling without replacement as if it were independent.
- Confidence intervals: Before constructing a confidence interval for a proportion or mean, you check that your sample meets the 10% condition so the standard error formulas are valid.
- Hypothesis tests: The same independence assumption underlies most hypothesis tests. Checking the 10% condition is a standard step before calculating test statistics.
In each case, you’re doing the same thing: confirming that your sample is small enough relative to the population that the slight dependency introduced by sampling without replacement won’t meaningfully distort your results.
A Different “Rule of 10” in Regression
If you came across the phrase “rule of 10” in the context of logistic regression, it refers to something entirely different. This guideline says you should have at least 10 outcome events for every predictor variable in your model. So if you’re predicting a rare disease and your model includes 5 risk factors, you’d want at least 50 cases of the disease in your data.
This rule originated from a 1996 simulation study that found serious problems when the ratio dropped below 10 events per variable: regression coefficients became biased, confidence intervals lost their accuracy, and the model sometimes found statistically significant results in the wrong direction. With 10 or more events per variable, these problems largely disappeared.
More recent research has suggested this threshold can sometimes be relaxed. A study in the American Journal of Epidemiology found acceptable performance with fewer than 10 events per variable under certain conditions, and noted that other factors like total sample size can be equally important. Still, the rule of 10 remains a widely taught starting point for deciding whether your dataset is large enough to support a logistic regression model.
When the 10% Rule Doesn’t Apply
The 10% rule only matters when you’re sampling without replacement from a finite population. If you’re sampling with replacement (putting each item back before drawing the next), your draws are already truly independent regardless of sample size. The rule also doesn’t apply when your population is effectively infinite, like measuring a manufacturing process that will produce millions of units.
If your sample does exceed 10% of the population, you’re not out of luck. You just can’t ignore the dependency. Instead, you apply the finite population correction factor directly, multiplying your standard error by the square root of (N − n) / (N − 1). This adjusts your calculations to account for the fact that you’ve sampled a substantial chunk of the population, which actually gives you more precise estimates than the simpler formula would suggest.