Confidence Interval Calculator

It is a range built from your sample using a procedure that, if repeated on many independent samples, would capture the true population value about 95% of the time. It is not the probability that the true value is inside this one specific interval; the 95% describes the long-run reliability of the method.

For a single mean, the calculator uses Student's t whenever you enter a sample standard deviation, regardless of sample size, with n - 1 degrees of freedom. A difference between two means uses Welch's t and Welch-Satterthwaite degrees of freedom when at least one sample standard deviation is greater than zero. If both are zero, the standard error is zero and the interval collapses to the observed difference, so the reported critical-value kind does not change the bounds. The proportion mode uses z inside the Wilson score interval because it is a binomial proportion rather than a sample-mean interval.

Margin of error is critical value times standard error, the half-width of the interval. A 95% CI of 100 plus or minus 4 has a margin of error of 4. Shrinking it requires a larger sample, a lower confidence level, or less variability.

Choose the confidence level before analyzing the data and follow the standard for your field. Higher confidence produces a wider interval for the same sample. A lower level produces a narrower interval but lower long-run coverage. Clinical and safety decisions usually require a full study design, not a confidence-level choice alone.

This calculator uses the Wilson score interval, which behaves better than the Wald interval near 0% and 100%. It still warns when np or n(1-p) is below about 10 because sparse endpoint data contain limited information. For formal work with very small samples, compare the Wilson result with an exact Clopper-Pearson interval.