A sample proportion has a distribution

If a categorical data set is modelled as a random sample from a categorical population, the sample proportions in the various categories must be treated as random quantities — they vary from sample to sample.

Sex of babies

Unknown probabilities

In some applications, we know the population probabilities for the categories of interest, but usually these values are unknown. (In practice, population parameters are usually unknown constants.) The corresponding sample proportions are approximations to these probabilities, but it is important to recognise that the underlying probabilities are unknown.

Effect of insecticide on beetles

Understanding the sample-to-sample variability of a proportion allow us to assess the proportion that is observed in a single observed data set.