Sampling Distribution of Proportion

1 min read Updated Fri Apr 24 2026 07:36:29 GMT+0000 (Coordinated Universal Time)

Assume a population of size $N$ with a known proportion of successes $p$ .

By drawing all possible samples of size $n$ and calculating their sample proportions $\overline{p}_i$ , the sampling distribution of the success proportion can be computed.

Mean

Average of all sample proportions $\overline{p}_i$ , denoted as $\overline{p}$ , is $p$ .

Standard Error

\sigma_p = \sqrt{\frac{p(1-p)}{n}}

Approx. to Normal Distribution

If the below conditions are met:

The sample size is large enough.
The population proportion is known.

Then:

\overline{p} \sim N \left(p,\frac{p(1-q)}{n} \right)