Standard Error of a Proportion Calculator
Find the standard error of a sample proportion from counts or a percentage, with finite population correction, margin of error and confidence interval.
The standard error of a proportion measures how much a sample percentage would typically change from one random sample to the next. Pollsters, quality-control teams and researchers use it to judge how precise a survey result or defect rate is. Enter the number of "successes" and the sample size, or a percentage, and the calculator returns the standard error, the margin of error and a confidence interval.
How to use this tool
- Choose whether you have a count of successes or a percentage.
- Enter the count (or percentage) and the sample size n.
- If the sample is a large share of a known population, enter the population size.
- Pick a confidence level and read the standard error, margin of error and interval.
Formula
SE = √( p̂ (1 − p̂) ÷ n ) with finite population correction: SE × √( (N − n) ÷ (N − 1) )
p̂ = x ÷ n is the sample proportion, n the sample size and N the population size. The margin of error is z × SE, where z is 1.645 for 90%, 1.960 for 95% and 2.576 for 99% confidence, and the interval is p̂ ± margin (the Wald interval). The correction assumes random sampling without replacement from that finite population; its effect grows with the sampling fraction.
Example
In a survey, 120 of 400 customers say they would recommend a product, so p̂ = 0.30. SE = √(0.30 × 0.70 ÷ 400) = √0.000525 = 0.022913, or 2.2913 percentage points. At 95% confidence the margin is 1.960 × 2.2913 = 4.4909 points, so the interval runs from 25.51% to 34.49%.
Frequently asked questions
When is this formula reliable?
The normal approximation behind it works when n·p̂ and n·(1 − p̂) are both at least about 10. For small samples, or proportions near 0% or 100%, use an exact (Clopper–Pearson) or Wilson interval instead; the calculator warns you when the rule is not met.
Why is the standard error largest at 50%?
p(1 − p) peaks at p = 0.5, so a 50/50 result is the least precise for a given sample size. That is why survey planners assume 50% when they have no earlier estimate.
How big a sample do I need for a ±3% margin?
At 95% confidence and p = 50%, n = (1.96 ÷ 0.03)² × 0.25 ≈ 1,068. Because SE shrinks with √n, quadrupling the sample only halves the standard error.
Is standard error the same as standard deviation?
No. The standard deviation of a yes/no variable is √(p(1 − p)) and describes individual answers. The standard error divides it by √n and describes the uncertainty of the sample proportion itself.
Last updated: 2026-10-30