Mathematics · Statistics
Confidence Interval Calculator — Mean or Proportion, Margin of Error
Find the confidence interval for a population mean from a sample's mean, standard deviation and size — using the t distribution, or z when the population SD is known — or for a proportion from the number of successes, with both the Wilson score and the simple normal (Wald) interval. Shows the margin of error and critical value.
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Quick start — try a typical case
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95% Confidence Interval
69.7255 to 75.2745
72.5 ± 2.77448
- Margin of Error
- ±2.77448
- Standard Error
- 1.36667
- Critical Value
- t = 2.0301 with 35 degrees of freedom
How this is calculated
Mean: x̄ ± t(α/2, n−1) × s ÷ √n (z if σ known) · Proportion (Wilson): (p̂ + z²/2n ± z√(p̂(1−p̂)/n + z²/4n²)) ÷ (1 + z²/n)
- α
- 1 − confidence level
A 95 % confidence interval is a range built so that, if you repeated the sampling many times, about 95 % of such ranges would contain the true value. It's the estimate plus or minus a margin of error: the critical value times the standard error. For small samples the t distribution's wider critical value allows for the extra uncertainty in s.
Assumptions & sources
- Random, independent sample; for means, a roughly normal population or n of about 30 or more.
- 1NIST/SEMATECH e-Handbook of Statistical Methods — US National Institute of Standards and Technology
- 2Probable inference, the law of succession, and statistical inference (Wilson score interval) — E. B. Wilson, Journal of the American Statistical Association, 1927
Calculator ID MATH-CI-001 · version 1.0.0
Common questions
What's the 95 % CI for a mean of 72.5, SD 8.2, n = 36?
72.5 ± 2.030 × 8.2/6 = 72.5 ± 2.77, i.e. 69.73 to 75.27 (t with 35 df).
What's the margin of error for a poll of 1,000 people?
At most about ±3.1 % at 95 % confidence (when the result is near 50 %).
Why use t instead of z?
When the standard deviation comes from the sample itself, t accounts for that extra uncertainty; with large n the two are almost the same.