Confidence Interval Calculator — From Summaries

Confidence Intervals. A confidence interval pairs an estimate with sampling uncertainty. Ninety-five percent describes the long-run coverage of the procedure, not a ninety-five percent probability about this fixed interval. Route the parameter first: an unknown-sigma t mean, a known-sigma z mean, a Wilson proportion, an exact binomial proportion, or a standard deviation. Choose confidence before looking at the endpoints. The remaining alpha is split across two tails, and the reference distribution supplies the critical quantile. For the usual mean interval, population sigma is unknown. Use x bar plus or minus t star times s over square root n, with n minus one degrees of freedom. Use the z branch only when population sigma is known independently. A large sample alone does not turn an estimated standard deviation into known sigma. For proportions, Wilson inverts the score test and behaves well near zero, one, and in smaller samples. Clopper-Pearson instead inverts exact binomial tails. Exact here means at-least-nominal coverage, not guaranteed equality. Those limits are often wider. For a standard deviation, a chi-square pivot creates asymmetric limits and assumes normality.

Start a ninety-five percent t interval with x bar ten, sample s two, and twenty-five independent observations. The standard error is point four. With twenty-four degrees of freedom, t star is about two point zero six three nine. The margin is point eight two five six, so the interval runs from nine point one seven four four to ten point eight two five six. Now observe fifty-four successes in eighty trials. The Wilson ninety-five percent interval is about fifty-six point six four to seventy-six point seven six percent. The exact binomial interval is about fifty-six point one one to seventy-seven point five five percent. Its extra width is the price of conservative tail coverage. With sample s two and n twenty-five, the normal-model interval for population standard deviation is one point five six one seven to two point seven eight two three. Asymmetry is correct. Margins shrink only with the square root of n, so halving one needs roughly four times the data. These tools do not repair clustering, selection bias, or nonnormal variance data. Build an interval free at Distri Scope dot com.