Sampling Distribution Sample Proportion Calculator

Sampling Distribution of a Sample Proportion

The sample proportion $\hat{p}$ estimates a population proportion $p$. Its sampling distribution is approximately normal when $np \ge 15$ and $n(1-p) \ge 15$. Related tools: P Hat Calculator and Sampling Distribution Probability Calculator.

Mean and Standard Error

$$\mu_{\hat{p}} = p$$

$$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$$

Probability Calculation

Convert a target proportion to a z-score with $Z = (p_1 - p)/\sigma_{\hat{p}}$, then use the normal CDF. For example, $P(\hat{p} > p_1) = 1 - \Phi(Z)$.

Example

With $p = 0.7$ and $n = 500$, the standard error is about 0.0205. Then $P(\hat{p} > 0.65) \approx 99.3\%$.

Frequently Asked Questions

When does the normal approximation fail?

When expected successes or failures are below 15, or when $p$ is very close to 0 or 1. Use exact binomial methods in those cases.

What is the difference between $p$ and $\hat{p}$?

$p$ is the population proportion. $\hat{p}$ is the proportion observed in a sample and varies from sample to sample.

Can I find $P(p_1 < \hat{p} < p_2)$?

Yes. Compute z-scores for both bounds and subtract cumulative normal probabilities.

How does sample size affect the distribution?

Larger $n$ shrinks standard error, so sample proportions cluster more tightly around $p$.

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