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The 10% condition (also called the 10 percent condition or the 10% rule in statistics) says that when you sample without replacement, the sample should be no more than 10% of the population, so the population is at least 10 times larger than the sample. Meeting it lets you treat the observations as approximately independent, which the standard formulas for proportions assume. This page shows the rule, where the AP Statistics framework requires it, and exact numbers for how much error 10% actually leaves behind.
What is the 10% condition?
When you draw individuals from a finite population without putting them back, each draw changes what is left, so the draws are not strictly independent. The 10% condition is a check that the sample is small enough, relative to the population, for that dependence to be ignorable. The College Board’s AP Statistics Course and Exam Description states it for sampling without replacement as: the population size must be at least 10 times larger than the sample size, N/n ≥ 10, where N is the population size and n is the sample size. It is a course convention, not a theorem: nothing special happens at exactly 10%, as the numbers below show.
10 percent condition formula (n ≤ 0.10 N)
Either of these equivalent forms is used:
- N / n ≥ 10
- n ≤ 0.10 × N
Symbols: N is the number of individuals in the population; n is the number of individuals in the sample. For two samples (a difference of two proportions), the College Board states it as n₁ ≤ 10% of N₁ and n₂ ≤ 10% of N₂: each sample is compared with its own population. If N is not known exactly, you only need to argue that the population is at least 10 times the sample size, for example that a school of several thousand students is far larger than a sample of 100.
Worked example: does the condition hold?
A school has N = 2,400 students. You survey a random sample without replacement.
- Sample of n = 150: N/n = 2,400 / 150 = 16, which is at least 10. Equivalently 0.10 × 2,400 = 240, and 150 ≤ 240. The condition is met.
- Sample of n = 300: N/n = 2,400 / 300 = 8, which is below 10. Equivalently 300 > 240. The condition is not met.
How much error does 10% leave? Exact numbers
Take a population of N = 1,000 in which exactly 500 individuals have a trait (p = 0.5). The count of individuals with the trait in a sample of size n follows a hypergeometric distribution when sampling without replacement, and is approximated by a binomial distribution when the 10% condition justifies independence (see binomial experiment). The standard deviation of the exact count is the binomial value multiplied by √((N − n)/(N − 1)). Computed values:
| Sample n | n / N | SD, binomial | SD, exact (hypergeometric) | Ratio exact / binomial | P(at least 60% with trait), binomial | Same, exact |
|---|---|---|---|---|---|---|
| 10 | 1% | 1.5811 | 1.5740 | 0.9955 | 0.37695 | 0.37633 |
| 50 | 5% | 3.5355 | 3.4477 | 0.9752 | 0.10132 | 0.09557 |
| 100 | 10% | 5.0000 | 4.7458 | 0.9492 | 0.02844 | 0.02238 |
| 200 | 20% | 7.0711 | 6.3277 | 0.8949 | 0.00284 | 0.00100 |
At a 10% sample the binomial standard deviation is about 5% too large, and a tail probability can be overstated by roughly a quarter (0.02844 versus 0.02238). At 20% the tail probability is almost three times too large. That is the practical reason for the 10% cutoff: below it the approximation error is modest, above it the independence assumption starts to distort results. The 10% threshold is a convenient convention, and the error grows gradually rather than jumping at 10%.
10% condition on AP Statistics: what the course framework requires
If you searched for the 10 condition in AP Stats, the College Board’s Course and Exam Description (checked 29 October 2026) uses it in these places:
| Procedure (CED topic) | What the CED says about the 10% condition |
|---|---|
| Sampling distribution of a sample proportion (3.2) | Sampling without replacement requires randomization and the 10% condition (N/n ≥ 10). |
| One-sample z-interval for a proportion (3.3) | Three conditions: randomization; 10% condition when sampling without replacement; at least 10 observed successes and 10 failures. |
| One-sample z-test for a proportion (3.5) | Same three, with the normality check using expected counts np₀ and n(1 − p₀), each at least 10. |
| Difference of two proportions (3.9) | Two independent random samples; n₁ ≤ 10% of N₁ and n₂ ≤ 10% of N₂. |
The course framework also lists the 10% condition among the checks for chi-square procedures, where the third condition is that expected counts are at least 5. The CED notes a common student error: incorrectly associating the 10% condition with the randomization condition. They are separate checks.
10% rule vs success-failure (np ≥ 10) condition
These are two different checks that happen to share the number 10. The 10% condition is about the sample size relative to the population and supports independence. The normality (success-failure) condition is about counts: expected or observed successes and failures each at least 10, which supports using a normal model for the sample proportion. Passing one says nothing about the other. For tests built on these conditions see population proportion and two-proportion z-test.
Assumptions and common mistakes
- Using it with replacement. The College Board states the condition for sampling without replacement. If units can be selected more than once, draws are independent and no check is needed. In the College Board’s wording, sampling without replacement means a unit can be selected only once.
- Treating it as proof of randomness. A small sample from a large population can still be a biased sample. See simple random sampling and sampling bias.
- Mixing it up with np ≥ 10. See the section above.
- Checking the wrong population in two-sample problems. Compare each sample with its own population.
- Treating 10% as a cliff. The table shows the error shrinking smoothly; a sample at 11% is not qualitatively different from one at 9%.
How to report it
State the check and the numbers in the methods or conditions paragraph. Example in APA style: “The 10% condition was met, because the sample (n = 150) was less than 10% of the population of 2,400 students (150 / 2,400 = 6.25%).” If the condition fails, say so and use a method that accounts for sampling from a finite population, such as the hypergeometric distribution.
How to calculate it in Excel, R or Python
Quick check of the condition itself: compute N / n and compare with 10. To see how close the binomial approximation is, compare tail probabilities for the example above (N = 1,000, 500 with the trait, n = 100, at least 60 with the trait).
Excel
=A1/B1 >= 10 (A1 = N, B1 = n) =1-HYPGEOM.DIST(59,100,500,1000,TRUE) exact, about 0.0224 =1-BINOM.DIST(59,100,0.5,TRUE) binomial, about 0.0284
R
N <- 1000; n <- 100 N / n >= 10 1 - phyper(59, 500, 500, 100) # exact 1 - pbinom(59, 100, 0.5) # binomial
Python
from scipy.stats import hypergeom, binom N, n = 1000, 100 print(N / n >= 10) print(hypergeom.sf(59, N, 500, n)) # 0.02238 print(binom.sf(59, 100, 0.5)) # 0.02844
In SciPy’s parameterisation, hypergeom(M, n, N) takes M total objects, n objects of the type of interest and N objects drawn, so hypergeom.sf(59, 1000, 500, 100) gives P(X > 59) for 100 draws from 1,000 objects of which 500 are of interest.
Frequently asked questions
What is the 10 percent condition in statistics?
It is the requirement that, when sampling without replacement, the population be at least 10 times larger than the sample (n ≤ 10% of N), so that the observations can be treated as approximately independent.
What is the 10 condition in AP Stats?
The same 10% condition: the College Board’s course framework requires it for sampling without replacement in proportion inference, alongside the randomization and normality conditions.
Is the 10 rule in statistics the same as np ≥ 10?
No. The 10% rule limits the sample relative to the population; np ≥ 10 and n(1 − p) ≥ 10 check that there are enough expected successes and failures for a normal model.
What if my sample is more than 10% of the population?
Independence is no longer a good approximation. The table above shows the binomial standard deviation overstating the exact value by about 5% at 10% and 12% at 20%. Use an exact finite-population method such as the hypergeometric distribution.
Do I need the 10% condition when sampling with replacement?
No; with replacement the draws are independent by construction.
Sources
- College Board, AP Statistics Course and Exam Description, topics 3.2, 3.3, 3.5 and 3.9 (date checked: 29 October 2026): https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf
- SciPy documentation,
scipy.stats.hypergeom(date checked: 29 October 2026): https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.hypergeom.html
The numerical table and worked examples were computed for this page from the hypergeometric and binomial probability mass functions.
