Margin of Error Calculator

Formula

How this is calculated

Select whether you are estimating a population mean or a population proportion, then pick a confidence level. Each confidence level maps to a critical z-value (z*): 90% → 1.645, 95% → 1.96, and 99% → 2.576. These come from the standard normal distribution and represent how many standard errors wide the interval must be.

For a mean, the standard error is σ/√n, so the margin of error is ME = z*·σ/√n, where σ is the population standard deviation and n is the sample size. For a proportion, the standard error is √(p̂(1−p̂)/n), giving ME = z*·√(p̂(1−p̂)/n), where p̂ is the sample proportion (between 0 and 1). The confidence interval is the point estimate ± ME — that is, x̄ ± ME for a mean or p̂ ± ME for a proportion.

This uses the normal (z) approximation, which assumes a reasonably large sample so the sampling distribution is approximately normal; for proportions the rule of thumb is that np̂ and n(1−p̂) should both exceed about 5–10. The margin of error shrinks with √n, so quadrupling the sample size halves the margin. Edge cases: n must be at least 1 (n ≤ 0 returns no result) and p̂ must lie between 0 and 1.