95% Confidence Interval Calculator
The 95% confidence interval for a mean — the most-quoted interval in science, with a plain statement of what it does and does not mean.
The interval is x̄ ± t × s/√n, where t is the 95% critical value on n − 1 degrees of freedom.
How the 95% confidence interval calculator works
The interval is x̄ ± t × s/√n, where t is the 95% critical value on n − 1 degrees of freedom. It is the range of population means that would not be rejected by a two-tailed test at 5%.
What it does not mean: there is not a 95% chance the true mean lies in *this* interval. The 95% refers to the procedure — across many repeated samples, 95% of the intervals it produces would contain the true mean.
Formula: CI = x̄ ± t₀.₉₇₅ × s / √n
Worked examples
| Inputs | Margin of error | Note |
|---|---|---|
| Mean 50, SD 10, n 40 | 3.198155 | ±3.20 |
| Four times the sample | 1.561372 | the margin roughly halves |
| A tiny sample | 15.912232 | the t value balloons to 3.18 |
FAQFrequently asked questions
What does a 95% confidence interval mean?
That the procedure produces intervals containing the true value 95% of the time. It is a statement about the method, not about this particular interval.
Is there a 95% chance the true mean is inside?
Not under the frequentist definition — the true mean either is or is not in it. The Bayesian credible interval is the one that supports that reading, and it needs a prior.
Why 95%?
Convention, traced back to Fisher's remark that two standard deviations was a convenient line. There is nothing special about it.
How do I make the interval narrower?
A bigger sample, mainly — the width falls as 1/√n. Reducing measurement noise also helps; lowering the confidence level narrows it only by weakening the claim.
What if two intervals overlap?
That does not mean the difference is non-significant. Overlapping intervals can still contain a significant difference; test the difference directly.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.