How this calculator works
The sample proportion p-hat is an estimate formed from the observed success count divided by the number of trials. Its complement is the observed failure proportion. The displayed standard error is the binomial plug-in estimate, which assumes independent trials with a common success probability. It is not a confidence interval and is misleading as a measure of certainty at all-success or no-success boundaries. A rounded percentage may not recover an exact integer sample count.
Formula
p-hat = x/n. q-hat = 1 - p-hat. Estimated standard error = sqrt[p-hat(1 - p-hat)/n]. Inverse forms: x = n × p-hat; n = x/p-hat.
Worked example
Thirty successes in 100 trials give p-hat = 0.30 and standard error sqrt(0.3 × 0.7/100) = 0.045826. A proportion of 40% in 50 trials corresponds to 20 successes.
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Frequently asked questions
Is p-hat the population probability?
It is a sample estimate. Sampling variation, nonresponse and selection bias can separate it from the population probability.
Why reject a fractional success count?
Observed successes count individual trials. A supplied percentage that implies 3.3 successes is not an exact count input.
Does zero standard error at the boundary prove certainty?
No. The plug-in estimate becomes zero at proportions zero and one. Interval methods designed for binomial counts handle that uncertainty more appropriately.