Sample proportion

Sample proportion (p̂) is the share of a sample with a given characteristic, calculated as p̂ = x ÷ n, used to estimate the true population proportion. Its variability is measured by its standard error.

ByHoang TruongUpdated July 11, 2026

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A quality check samples 250 packages and finds 15 damaged, giving a sample proportion p̂ of 15 divided by 250, or 6%. Its standard error is the square root of 6% times 94% divided by 250, about 1.5 percentage points, which is how much this estimate could plausibly vary from another sample of the same size.

Where it fits

SubjectData Analysis & StatisticsCoreTopicEstimation & Sampling DistributionsCoreTopicCommon Significance TestsCore

The formula

LaTeX

\hat p = \dfrac{x}{n}

Variables

$\overset{p}{^}$: Sample proportion
$x$: Number of sample members with the characteristic (count)
$n$: Sample size (count)

Estimates the population proportion from the share of the sample having the characteristic of interest.

LaTeX

SE(\hat p) = \sqrt{\dfrac{\hat p (1-\hat p)}{n}}

Variables

$\overset{p}{^}$: Sample proportion
$n$: Sample size (count)

Measures how much the sample proportion is expected to vary from sample to sample of the same size, the basis for confidence intervals and z-tests on proportions.

Check yourself

PracticeCORE

A quality team samples n = 400 invoices and finds that x = 88 contain at least one error. What is the standard error of the sample proportion of invoices with errors?

Select an answer to check your understanding.