Confidence Interval Calculator

Take the sample mean and add and subtract the margin of error: mean ± (critical value × standard error), where the standard error is the standard deviation divided by the square root of the sample size. Use a z critical value when the population standard deviation is known, and a t critical value (with n − 1 degrees of freedom) when it is estimated from the sample.

Use z when you know the true population standard deviation (rare) or your sample is very large. Use t when you estimate the standard deviation from the sample, which is the usual case. The t-distribution has wider tails to account for the extra uncertainty, and it converges to z as the sample grows.

Use p̂ ± z × √(p̂(1 − p̂) / n), where p̂ is the sample proportion and z is the critical value for your confidence level (1.96 for 95%). This is the standard Wald interval and works well when both n·p̂ and n·(1 − p̂) are at least about 10.

It means that if you repeated the sampling process many times and built an interval each time, about 95% of those intervals would contain the true value. It does not mean there is a 95% probability that this particular interval contains the true value — the true value is fixed, the interval is what varies.

The margin of error is the half-width of the confidence interval: the critical value times the standard error. A larger sample, a smaller standard deviation, or a lower confidence level all shrink the margin of error and produce a tighter interval.

Z critical values come from an inverse-normal (probit) approximation accurate to about 1e-9, and t critical values are found by inverting the Student-t CDF (built on the regularized incomplete beta function). Both match standard statistical tables.