populationsproportion

Populationsproportion, also called population proportion, is the proportion of individuals in a population that possess a given attribute. It is a population parameter denoted by p and lies between 0 and 1.

When sampling, the number X of successes in a sample of size n follows a Binomial(n, p). The sample proportion p_hat = X/n is an unbiased estimator of p with variance p(1-p)/n. The standard error is sqrt(p(1-p)/n); in practice, p_hat is used in place of p for the standard error.

For inference, p_hat is approximately normally distributed in large samples. Confidence intervals include the Wald interval p_hat ± z_(α/2) sqrt(p_hat(1-p_hat)/n), though Wilson score and Agresti-Coull intervals often perform better, particularly for small samples or extreme p values. Hypothesis testing about p uses a z-test for large samples or an exact binomial test for small samples.

In epidemiology and public health, the population proportion corresponds to prevalence—the proportion of individuals with a condition at a given time. In survey research, estimating p requires consideration of the sampling design, including weights and the finite population correction when a substantial fraction of the population is sampled. Limitations include potential bias from non-representative samples, small-sample inaccuracies of normal-based methods, and the need to account for complex sampling in variance estimation.