Sample Proportion Calculator
The proportion in a sample, its standard error and a confidence interval — with the Wilson interval, which behaves properly when the count is small.
The sample proportion p̂ = x ÷ n estimates the population proportion.
How the sample proportion calculator works
The sample proportion p̂ = x ÷ n estimates the population proportion. Its standard error is √(p̂(1 − p̂)/n), and the familiar interval is p̂ ± z × SE.
That familiar interval fails badly near 0 and 1 — it can even run below zero. The Wilson score interval solves it by inverting the test rather than approximating it, and is what the calculator leads with for small samples.
Formula: p̂ = x/n; SE = √(p̂(1 − p̂)/n)
Worked examples
| Inputs | Sample proportion | Note |
|---|---|---|
| 45 of 200 | 22.5% | 22.5%, ±5.8 points |
| A clean run: 0 of 30 | 0% | Wilson still gives a sensible upper bound |
| 95 of 100 | 95% | 95%, and the normal interval overshoots |
FAQFrequently asked questions
What is a sample proportion?
The share of a sample with some property — 45 of 200 is 22.5%.
Why is the Wilson interval better?
The usual normal interval assumes symmetry that does not hold near 0 or 1. At 0 successes it gives a nonsensical interval of exactly [0, 0]; Wilson gives a proper upper bound.
What is the standard error?
The typical distance between the sample proportion and the true one — √(p(1−p)/n). It shrinks with the square root of n, so quadrupling the sample halves it.
How many do I need for ±3 points?
About 1,067 at 95% confidence, for a proportion near 50%.
What if I saw zero successes?
The rule of three gives a rough upper bound of 3/n. With 30 trials and no events, the true rate could still be as high as about 10%.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.