How this calculator works
For a mean, divide the SD by sqrt(n), then multiply by a two-sided critical value. Student t uses n - 1 degrees of freedom when the SD is estimated from the sample. The normal method requires a known population SD. Proportion mode uses the Wilson score interval, which remains within [0, 1] even at zero or all successes. All methods assume an appropriate random sampling design.
Formula
Mean CI = x̄ ± critical × SD/√n. Wilson centre = (p̂+z²/(2n))/(1+z²/n); half-width = z√(p̂(1-p̂)/n+z²/(4n²))/(1+z²/n).
Worked example
A mean of 100, known SD 15 and n = 25 give SE 3. A normal 95% interval is 100 ± 1.959964 × 3, or [94.120108, 105.879892].
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Frequently asked questions
Is there a 95% probability that this fixed parameter lies inside?
Frequentist coverage refers to the method over repeated samples, with the parameter fixed.
When does Student t work well?
For independent normal observations it has exact coverage. With other distributions, sample size and departure from normality matter.
Why use Wilson for proportions?
It avoids the impossible negative or above-one endpoints that simple Wald intervals can produce.