Confidence Interval Calculator

How to use the calculator

1. Choose the mode: mean (a measurement such as a score or a weight) or proportion (a share of yes answers).
2. Pick the confidence level. 95% is the most common choice.
3. Mean mode: enter the sample mean, the standard deviation and n. Keep "sample standard deviation" unless σ is genuinely known from outside the sample.
4. Proportion mode: enter the number of successes and the sample size.
5. Mean mode shows the bounds, the margin of error, the standard error and, with s, the degrees of freedom. Proportion mode shows the Wilson bounds, the sample proportion, the Wald interval and margin, and the Wilson center. The table repeats the interval at every level; in proportion mode its bounds are Wilson and its margin is Wald.

Formulas used

Mean

Interval = x̄ ± c × s / √n, where s / √n is the standard error. With a sample standard deviation, c is the Student's t critical value with n − 1 degrees of freedom, so you need at least 2 observations. With a known population σ, c is the normal value z, the formula given in the NIST/SEMATECH e-Handbook. The t quantiles are computed numerically and match the NIST t table, for example 2.228 for 10 degrees of freedom at 95%.

Proportion

• Wald: p̂ ± z × √(p̂(1 − p̂) / n), with p̂ = successes / n, clipped at 0% and 100%. NIST calls it the expression most frequently used and notes that its lower limit can be negative.
• Wilson score (the main result): centered on (p̂ + z²/2n) / (1 + z²/n) rather than on p̂. NIST notes that Brown, Cai and DasGupta recommend it, and that Agresti and Coull recommend it for virtually all combinations of n and p.

Confidence interval examples

1. Mean 72.4, s = 9.8, n = 100, 95%

t = 1.9842 (99 degrees of freedom), standard error 0.98, margin ±1.9445, interval 70.4555 to 74.3445. If 9.8 were a known σ, z = 1.959964 gives 70.4792 to 74.3208, slightly narrower.

At 99% the n = 100 margin grows to ±2.5739; at 80% it shrinks to ±1.2644.

2. 62 successes out of 100, 95%

Wilson: 52.21% to 70.90%, centered at 61.56%, so it is not symmetric around p̂ = 62%. Wald: 52.49% to 71.51%, ±9.51 points.

3. Rare events: 1 out of 20

Wald gives 0% to 14.55% after clipping a negative lower bound. Wilson gives 0.89% to 23.61%. With 0 out of 20, Wald collapses to 0% to 0%, an interval of zero width, while Wilson still gives 0% to 16.11%. These are the textbook reasons to prefer Wilson.

Check against NIST: 26 out of 200 at 90% gives a Wilson lower bound of 9.577%, the 0.09577 in the e-Handbook's example. NIST uses it as a one-sided 95% bound, which takes the same z as a two-sided 90% interval.

What a 95% confidence interval means

The 95% describes the method, not one particular interval. In NIST's words, if the same population were sampled on numerous occasions, the intervals would bracket the true value in approximately 95% of the cases. It does not mean there is a 95% probability that the true mean lies in the interval you just computed: that interval either contains it or it doesn't.

• The margin covers random sampling error only. A biased, non-random or self-selected sample, or one with heavy non-response, can be far off with a narrow interval.
• Not modeled: finite-population correction, one-sided intervals, differences between two means or proportions, exact (Clopper–Pearson) and bootstrap intervals.