Confidence Interval Calculator
Get a confidence interval for a mean from the sample mean, standard deviation and sample size, using the t distribution or z.
Last updated: October 2026
Confidence interval (lower to upper)
71.40 to 84.60
Formula below · 2 sources (itl.nist.gov, Wikipedia) · Updated Oct 2026
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About this calculator
A confidence interval gives a range of plausible values for a population mean, based on a sample.
CI = x̄ ± critical value × SE, with SE = s ÷ √n
x̄ is the sample mean, s the sample standard deviation, n the sample size and SE the standard error of the mean. If you already have the standard error (from software or a paper), enter it in the optional field: it replaces s ÷ √n and the z critical value is used, since the sample size behind it may not be known.
t or z?
When the standard deviation comes from the sample itself, which is almost always, the critical value comes from the t distribution with n − 1 degrees of freedom. The calculator computes it exactly: for 95% confidence it is 2.064 at n = 25, 2.228 at n = 11 and 1.984 at n = 101, approaching the z value of 1.96 as n grows. Choose z only when the population standard deviation is known or the sample is very large; with small samples z gives an interval that is too narrow.
What 95% means
If you repeated the sampling many times, about 95% of the intervals built this way would contain the true mean. The interval assumes a random sample and a roughly normal sampling distribution, which holds for normal data or, by the central limit theorem, for moderately large samples. It covers sampling error only, not bias from how the sample was collected. The chart under the result draws the interval when a standard error is entered.
How to use
Enter the sample mean, the sample standard deviation and the sample size, then pick the confidence level. The default, a mean of 78 with s = 16 from 25 people at 95%, has SE = 16 ÷ √25 = 3.2. The t critical value for 24 degrees of freedom is 2.064, so the margin is 2.064 × 3.2 = 6.60 and the interval is 71.40 to 84.60. Switch to z and the margin becomes 1.96 × 3.2 = 6.27, giving 71.73 to 84.27. With only 11 people and the same s, SE = 4.82, t = 2.228 and the 95% interval widens to 67.25 to 88.75. To use a known standard error instead, type it into the last field: a mean of 25.5 with SE 0.8 at 99% gives 25.5 ± 2.576 × 0.8 = 23.44 to 27.56.
Frequently asked questions
Does a 95% confidence interval mean a 95% chance the true mean is inside it?
Strictly, no. The 95% describes the method: across many samples, 95% of intervals built this way would contain the true mean. Any one interval either contains it or not. In practice the interval is read as the range of values consistent with the data.
Why is the t interval wider than the z interval?
Using s in place of the unknown population standard deviation adds uncertainty, and the t distribution has heavier tails to account for it. The difference is large for small samples (t = 2.776 at 4 degrees of freedom versus z = 1.96) and small once n is above about 30.
How does sample size affect the width?
The width shrinks with the square root of n. Quadrupling the sample roughly halves the interval; doubling it narrows the interval by about 29%.
Can I use this for a percentage or proportion?
No. For a proportion, such as the share of voters who back a candidate, use the margin of error calculator, which uses the standard error of a proportion, √(p(1 − p) ÷ n).
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