🎯Confidence Interval Calculator

About this tool

Choose an interval for a mean or for a proportion, and a confidence level from 80% to 99.9%. For a mean, enter the sample mean, standard deviation and sample size, and tick the box if the population standard deviation is known. For a proportion, enter the number of successes and the sample size. You get the interval, the margin of error, the standard error and the critical value used.

How confidence intervals work

A confidence interval is a range of plausible values for a population figure, worked out from a sample. For a mean it is x̄ ± critical value × standard error, where the standard error is s ÷ √n. When the population standard deviation is unknown, which is usual, the critical value comes from Student's t distribution with n − 1 degrees of freedom. That is wider than the normal value for small samples: t = 2.023 for 40 observations at 95%, against z = 1.960. A sample of 40 with a mean of 72.4 and a standard deviation of 8.5 gives a 95% interval of about 69.7 to 75.1. For proportions, the Wilson score interval is recommended because it behaves well near 0% and 100% and for small samples. 312 successes out of 800 is 39%, with a 95% Wilson interval of about 35.7% to 42.4%. The meaning of 95% confidence is subtle. The method captures the true value in 95% of repeated samples, which is not quite the same as a 95% chance that this particular interval contains it. Honest limits: the formulas assume a random sample, and for small samples of means, roughly normal data. The sample size calculator plans how many to survey, the z-score calculator converts critical values and the A/B test calculator compares two groups. Standard deviation explained: what it measures, how it's calculated, and the 68–95–99.7 rule explains the spread behind the standard error.

Frequently asked questions