What Is the Sampling Distribution of a Sample Proportion?
When you draw random samples of size n from a population with proportion p, the sample proportion varies from sample to sample. The distribution of all possible sample proportions is called the sampling distribution of the sample proportion. By the Central Limit Theorem, this distribution is approximately normal when the sample size is large enough.
The mean of the sampling distribution equals the population proportion p, and the standard deviation (called the standard error) equals the square root of p(1-p)/n. This relationship is fundamental to constructing confidence intervals and performing hypothesis tests for proportions in survey research, quality control, and clinical trials.
Formulas
Conditions for Normal Approximation
| Condition | Requirement |
|---|---|
| Success condition | np ≥ 10 |
| Failure condition | n(1-p) ≥ 10 |
| Independence | n ≤ 10% of population (if sampling without replacement) |
Frequently Asked Questions
Why must np and n(1-p) both be at least 10?
These conditions ensure the sampling distribution is sufficiently bell-shaped for the normal approximation to be accurate. When either condition fails, the distribution is too skewed, and exact binomial methods should be used instead.
How does sample size affect the standard error?
Increasing sample size decreases the standard error proportionally to the square root of n. Quadrupling the sample size halves the standard error. This is why larger samples give more precise estimates of the population proportion.