90% Confidence Interval Calculator
The 90% confidence interval for a mean — narrower than the usual 95%, and standard in one-sided settings and equivalence testing.
Same construction, a smaller critical value: 1.
How the 90% confidence interval calculator works
Same construction, a smaller critical value: 1.645 rather than 1.960 in a large sample, so the interval is about 16% narrower than the 95% one.
It appears more often than its reputation suggests. A two-sided 90% interval is exactly the region a one-sided 5% test would not reject, which is why regulators use it for bioequivalence.
Formula: CI = x̄ ± t₀.₉₅ × s / √n
Worked examples
| Inputs | Margin of error | Note |
|---|---|---|
| Mean 50, SD 10, n 40 | 2.664021 | ±2.66 |
| A large sample | 0.232661 | t approaches 1.645 |
| Zero spread | 0 | a point, not an interval |
FAQFrequently asked questions
When is a 90% interval appropriate?
Exploratory work, and any setting where the question is genuinely one-sided. Bioequivalence testing uses a two-sided 90% interval by regulation.
Is it less rigorous?
It is a weaker claim from the same data, which may be exactly the right claim. What is not acceptable is choosing 90% after 95% failed.
How much narrower is it?
About 16% in a large sample.
Why does a 90% two-sided interval match a 5% one-sided test?
Because each tail of a two-sided 90% interval holds 5%, which is precisely the rejection region of a one-sided test at that level.
Which should I report?
Whichever you specified before analysing. State the level explicitly — an unlabelled interval is unreadable.
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)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.