Margin of Error Calculator

Margin of error is a confidence-bound around a sample estimate. A poll with a margin of error of plus or minus 3 percent at 95 percent confidence says, in plain terms, that if the poll were repeated many times under the same conditions, about 95 percent of those polls would land within 3 percentage points of the true population value. It does not mean every poll is within that range, and it ignores non-sampling errors like wording bias or response refusal.

First compute the standard error: the square root of p × (1 − p) divided by sample size, where p is the observed proportion. Then multiply by the z-score for your confidence level (1.645 for 90 percent, 1.96 for 95 percent, 2.576 for 99 percent). The product is the margin of error in decimal form; multiply by 100 to express as a percent.

Sample variance p × (1 − p) is largest when p = 0.5, which makes the margin of error largest at that point. Using 50 percent gives the most conservative (worst-case) margin of error, which is the standard convention when the true proportion is unknown ahead of polling.

When sample size is a meaningful fraction of the total population, the standard formula overstates the margin of error. The finite-population correction (FPC) multiplies the margin by the square root of (N − n) divided by (N − 1), where N is the total population and n is the sample size. It only changes the result meaningfully when the sample is more than about 5 percent of the population.

Margin of error shrinks as sample size grows, but the relationship is not linear. Doubling the sample size shrinks the margin by a factor of roughly the square root of 2, or about 30 percent. Going from 400 to 1,600 respondents cuts the margin roughly in half. Once samples are large (say, several thousand), the marginal value of adding more drops off.

No. This calculator is for planning, education, and quick sanity checks. Real survey design has to consider question wording, sampling frame, response rate, weighting, design effect from clustered or stratified samples, and non-sampling error. For formal research use professional statistical software or consult a qualified statistician.