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Sampling distribution of the sample proportion calculator

Model repeated sample proportions from a known population proportion. Inspect expected success and failure counts before trusting the normal approximation.

Mean sample proportion
50.000000%
Standard error
0.050000
Approximate probability
97.724987%
Expected successes
50.000000
Expected failures
50.000000
Normal approximation check
Both expected counts are at least 10; also check independence and sampling design.

Estimates for information only. Not financial, medical, legal or engineering advice. See the Terms.

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How this calculator works

The sampling mean is p and its SD is sqrt(p(1-p)/n). Without replacement, multiply variance by (N-n)/(N-1). Probabilities use a normal approximation without a continuity correction. At the physical limits, probabilities are bounded by the proportion support [0, 1]; use binomial or hypergeometric counts for exact results, especially in the tails. A census or p equal to zero or one gives a certain result.

Formula

E[p̂] = p; SE = √(p(1-p)/n); finite-population SE = SE√((N-n)/(N-1)); z = (p̂-p)/SE.

Worked example

For p = 0.5 and n = 100, SE = 0.05. The uncorrected normal chance below 0.60 is Φ(2) = 0.9772499.

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Frequently asked questions

Does this estimate p from observed successes?

No. It assumes a population p to describe repeated samples. Use a confidence interval to estimate an unknown p.

Why check expected counts?

A normal curve can poorly describe a discrete, skewed sampling distribution near the boundaries.

When should I use finite correction?

For a simple random sample without replacement from a known finite population.

Sources

Written byAmira MansourCalculator writer. Builds and tests Mathulate calculators against published sources.
Reviewed byMarc LefebvreMathematics teacher. Checks each formula and worked example by hand.

Last updated .

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