Confidence Interval Calculator

When you measure a sample instead of an entire population, the number you get is only an estimate. A confidence interval wraps that estimate in a range of plausible values for the true population parameter, such as a mean or a proportion. Reporting the interval rather than a single point tells your audience how precise the estimate actually is and how much sampling variation it carries.

The math behind the interval follows a simple structure: take your estimate and add and subtract a margin of error, written as Estimate ± (Critical Value × Standard Error). The estimate is the figure straight from your sample, such as the sample mean. Because the standard deviation you enter is estimated from the sample itself, the calculator uses a t critical value with n − 1 degrees of freedom for the confidence level you choose, as the NIST handbook does. For small samples it is noticeably larger than the normal-distribution z value (2.262 rather than 1.96 at 95% with n = 10, and 2.776 with n = 5); as the sample grows it approaches the z values of 1.645, 1.96 and 2.576 for 90%, 95% and 99%. The standard error measures how much the sample estimate would bounce around from one sample to the next.

The confidence level is the part people most often read wrong. A 95% confidence interval does not say there is a 95% chance the true value sits inside your particular interval. It says that if you drew sample after sample and built an interval each time, 95% of those intervals would capture the true population parameter. The interpretation only holds when the sample is drawn randomly and represents the population, so a biased sampling method quietly breaks the whole result.