Confidence Interval Calculator

A confidence interval gives a range of plausible values for something you have estimated from a sample, such as an average or a percentage. This calculator builds intervals for a single mean (with the t or z distribution), for a proportion (Wilson, Wald and exact Clopper–Pearson) and for the difference between two group means (Welch's method), from summary numbers or from your raw data.

How to use this tool

1. Choose what the interval is for: a mean, a proportion or a difference of two means.
2. Pick the confidence level; 95% is the usual choice.
3. Enter the summary values (mean, standard deviation and sample size, or successes and trials), or paste raw data.
4. Read the interval, the margin of error and the critical value used.

Formula

Mean: x̄ ± t* × s/√n · Proportion (Wilson): [p̂ + z²/2n ± z√(p̂(1−p̂)/n + z²/4n²)] ÷ (1 + z²/n) · Two means: (x̄₁ − x̄₂) ± t* √(s₁²/n₁ + s₂²/n₂)

t* comes from the t distribution with n − 1 degrees of freedom (Welch–Satterthwaite degrees of freedom for two groups); use z* when the population standard deviation is truly known. The Clopper–Pearson limits are the beta-distribution quantiles that make the binomial tail probabilities equal to α/2.

Example

A sample of 25 has mean 50 and SD 10. The standard error is 10 ÷ √25 = 2, and t* for 24 degrees of freedom at 95% is 2.0639, so the interval is 50 ± 4.13 = 45.87 to 54.13. For 40 successes out of 100, the Wilson 95% interval is 30.9% to 49.8%, slightly shifted toward 50% compared with the simple 30.4–49.6%.