Confidence Interval Calculator
Choose a mean (with the population standard deviation, or the sample standard deviation) or a proportion. Enter your sample and a confidence level to get the lower and upper bounds.
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Formula
- Mean, σ known:
mean ± z × σ / √n - Mean, σ unknown:
mean ± t × s / √n, withn - 1degrees of freedom - Proportion (Wilson score), with
p = successes / n:centre = (p + z²/(2n)) / (1 + z²/n)half-width = z / (1 + z²/n) × √(p(1-p)/n + z²/(4n²))- interval =
centre ± half-width, kept within 0 and 1
The critical value z (or t) is the 1 - (1 - level)/2 quantile: z = 1.645 (90%), 1.960 (95%), 2.576 (99%). A custom level accepts 50 to 99.999.
Worked example
Mean 50, s = 10, n = 25, 95% confidence, σ unknown:
- Standard error = 10 / √25 = 2
- t(24, 0.975) = 2.0639
- Margin = 2.0639 × 2 = 4.1278
- Interval = 45.8722 to 54.1278
With σ = 10 known, z = 1.96 gives a margin of 3.9199 and an interval of 46.0801 to 53.9199. For the Wilson interval, 20 of 100 at 95% gives 0.1334 to 0.2888.
Frequently asked questions
How do you calculate a confidence interval for a mean?
Use mean ± critical value × s / √n. With mean 50, s = 10, n = 25 and 95% confidence, t(24) = 2.0639, so the margin is 2.0639 × 10 / 5 = 4.1278 and the interval is 45.87 to 54.13.
When should I use z and when t?
Use z only when the population standard deviation σ is known. If you estimate it from the sample (the usual case) use t with n - 1 degrees of freedom. For the same data with σ = 10 known, the margin is 1.96 × 2 = 3.92 instead of 4.13.
What is the Wilson interval for a proportion?
The Wilson score interval is a more reliable alternative to p ± z√(p(1-p)/n), especially for small n or p near 0 or 1. For 20 successes in 100 at 95% it is 0.1334 to 0.2888, i.e. 13.3% to 28.9%.
What does 95% confidence really mean?
If you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true value. It is not a 95% probability that this one interval does.
How does the confidence level change the width?
Higher confidence needs a larger critical value, so a wider interval. The z values are 1.645 for 90%, 1.960 for 95% and 2.576 for 99%, so going from 95% to 99% widens the interval by about 31%.