10% Condition — Definition, Formula & Examples
The 10% Condition states that when sampling without replacement, the sample size must be no more than 10% of the population size. This condition allows you to treat individual observations as approximately independent, even though they technically are not.
When drawing a sample of size without replacement from a finite population of size , the observations may be treated as independent provided . This ensures that the probability of each outcome changes negligibly from one draw to the next, so formulas that assume independence (such as those for binomial probabilities or standard errors) remain valid approximations.
Key Formula
Where:
- = Sample size
- = Population size
How It Works
Before performing inference—whether constructing a confidence interval or running a hypothesis test—check that your sample is small relative to the population. Divide the population size by 10 and confirm your sample size does not exceed that value. If the condition is met, the slight dependence introduced by sampling without replacement is negligible, and you can proceed using standard formulas. If it is violated, the standard error formulas will overestimate the true variability, and a finite population correction factor is needed.
Worked Example
Problem: A quality control inspector selects 80 light bulbs from a shipment of 1,200 to test for defects. Does the sample satisfy the 10% Condition?
Calculate 10% of the population: Find 10% of the population size.
Compare sample size to the threshold: Check whether the sample size is at most 120.
Answer: Yes, 80 is less than 120, so the 10% Condition is satisfied. The inspector can treat the sampled bulbs as independent observations.
Why It Matters
On the AP Statistics exam, you must verify the 10% Condition whenever you check conditions for inference involving proportions or means from a finite population. Failing to state it can cost you points on free-response questions, even if your calculations are otherwise correct.
Common Mistakes
Mistake: Confusing the 10% Condition with the Large Counts Condition and thinking they check the same thing.
Correction: The 10% Condition checks whether observations are approximately independent (sample size vs. population size). The Large Counts Condition checks whether the sampling distribution is approximately normal (using and ). Both must be verified separately.