General notation

In a categorical population, we choose one category of interest and call it success; all other categories are collectively called failures. The population proportion of successes is denoted by π.

When a random sample of n values is selected, we denote the number of successes by x and the proportion of successes by p  = x/n.

Distribution of a sample proportion

The number of successes, x , has a 'standard' discrete distribution called a binomial distribution which has two parameters, n and π.

In practical applications, n is a known constant, but π may be unknown. The sample proportion, p , has a distribution with the same shape, but is scaled by n .