Confidence Interval Calculator

A confidence interval is a range of plausible values for a population parameter, built from sample data. A 95% confidence interval for the mean is x̄ ± (critical value) × s/√n.

Use z when the population standard deviation σ is known or the sample is large (n ≥ 30). Use t with df = n − 1 when σ is estimated from the sample, especially for small samples — t has heavier tails, so its intervals are wider.

Not exactly. It means that if you repeated the sampling process many times, about 95% of the intervals you build would contain the true mean. The true mean is fixed; the interval is what varies.

A bigger n shrinks the standard error s/√n, so the margin of error shrinks and the interval gets narrower — you learn the mean more precisely.

Higher confidence demands a larger critical value (2.576 instead of 1.96), which multiplies the standard error into a wider margin of error. More certainty costs more width.