Confidence interval calculator

If you repeated the sampling 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% chance the true value is in this particular interval; the true value is fixed, and the interval either contains it or not.

Use t when you estimate σ from the sample, which is almost always. Use z only when σ is known from long history. For large samples the two give nearly the same interval.

The Wald interval p̂ ± z√(p̂(1 − p̂)/n) is the textbook formula, but it is unreliable when np or n(1 − p) is under about 5 and can run below 0. The Wilson interval behaves well for small samples and proportions near 0 or 1, and is the better choice in practice.

The chi-square distribution is skewed, so the interval for the variance, and for σ, is longer on the high side. That is also why you need two different chi-square values.

Increase the sample size (the width shrinks with √n), or accept a lower confidence level. Reducing process variation also narrows it.

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