Sampling Distribution of Proportion

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.

Note

Distribution of proportions can be modelled using a binomial distribution.

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:

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