A 90% confidence interval trades a little certainty for a narrower, more precise range. It leaves 5% of probability in each tail of the sampling distribution, so its normal critical value is z* = 1.645 rather than the 1.96 used at 95%. The interval is therefore about 16% narrower than the 95% interval from the same data.
Ninety percent intervals are common in quality control, engineering tolerances, pilot studies and business forecasting, where decisions are made often and a slightly higher miss rate is acceptable. They also have a neat link to testing: a two-sided 90% interval corresponds to a one-sided test at the 5% level, so it is the interval used in equivalence testing (the “two one-sided tests” approach) for comparing drugs or processes.
Enter your summary statistics; the confidence level is preset to 90% but you can change it. When the standard deviation comes from the sample itself the calculator uses the t distribution, whose 90% critical value is a little larger than 1.645 for small samples (1.833 with 9 degrees of freedom, 1.690 with 35).
How to use the 90% Confidence Interval Calculator
- Calculate a confidence interval for
- Sample mean (x̄)
- Spread you know
- Standard deviation or standard error
- Sample proportion (p̂)
- Sample size (n)
- Confidence level
- Custom confidence level
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Compute the standard error s ÷ √n, multiply it by the 90% critical value (z* = 1.645, or t* with n − 1 degrees of freedom when s is a sample statistic) and add the result to and subtract it from the sample mean.
Worked example
A filling machine is checked by weighing 36 cereal boxes: mean 502.4 g, s = 4.6 g. The standard error is 4.6 ÷ 6 = 0.767 g. With 35 degrees of freedom the 90% critical value is t* = 1.6896, so the margin of error is 1.295 g and the 90% confidence interval is 501.10 g to 503.70 g. The whole interval lies above the 500 g label weight. With the normal value 1.645 instead, the interval would be 501.14 to 503.66 g.
90% compared with other levels
| Level | z* | Width relative to 95% |
|---|---|---|
| 80% | 1.2816 | 65% |
| 90% | 1.6449 | 84% |
| 95% | 1.9600 | 100% |
| 99% | 2.5758 | 131% |
Formula
90% CI = x̄ ± 1.645 × σ ÷ √n (σ known) · x̄ ± t*₀.₉₅,ₙ₋₁ × s ÷ √n (s from the sample) · α = 0.10, 5% in each tail
Worked example
Inputs — Calculate a confidence interval for: A population mean; Sample mean (x̄): 502.4; Spread you know: Sample standard deviation s (t interval); Standard deviation or standard error: 4.6; Sample size (n): 36; Confidence level: 90%.
Results — 90% confidence interval = 501.1047 to 503.6953 (502.4 ± 1.2953); Lower bound = 501.1047; Upper bound = 503.6953; Margin of error (±) = 1.2953; Standard error = 0.7667; Critical value = 1.6896.
Key terms
- 90% confidence
- Ten percent of intervals built this way miss the true value, 5% on each side.
- z* = 1.645
- The 95th percentile of the standard normal distribution.
- Equivalence test
- A test that two treatments differ by less than a set margin, often based on a 90% interval.
Frequently asked questions
What is the z-score for a 90% confidence interval?
z* = 1.645 (more precisely 1.6449), because 5% of the standard normal distribution lies above it and 5% below −1.645.
How do I calculate a 90% confidence interval?
x̄ ± 1.645 × σ ÷ √n. For x̄ = 50, σ = 10, n = 25: 50 ± 1.645 × 2 = 46.71 to 53.29.
Is a 90% interval narrower than a 95% interval?
Yes. Its critical value is 1.645 instead of 1.96, so it is about 16% narrower for the same data.
When should I use 90% confidence?
In exploratory work, quality control, forecasting and equivalence testing, where a 10% chance of missing the true value is acceptable.
What t value is used for 90% with a small sample?
t* with n − 1 degrees of freedom at the 95th percentile, e.g. 1.833 for n = 10 and 1.699 for n = 30.
How does a 90% interval relate to hypothesis tests?
A value outside the two-sided 90% interval would be rejected by a two-sided test at α = 0.10, or a one-sided test at α = 0.05 in that direction.