How this instrument works
A confidence interval is a range built around a sample mean that is likely to contain the true population mean. '90% confidence' describes the long-run behaviour of the method, not any single interval: if you repeated the same sampling process many times and built a 90% interval each time, about 90% of those intervals would capture the true mean. This instrument fixes the confidence level at 90% and the normal critical value at z = 1.6449, so every result uses exactly the same standard.
Ninety percent is the narrowest of the three critical values this site offers as dedicated instruments (90%, 95%, 99%), because a lower confidence level buys a tighter range in exchange for a slightly higher chance of missing the true mean. That trade-off makes it a natural fit for quick, lower-stakes looks — early A/B test reads, exploratory business analytics, or quality-control screening — where a narrower band is more useful for spotting a signal than the extra certainty of a wider one.
The interval is built from three ingredients: the sample mean x-bar (your point estimate), the standard error s/√n (how much that estimate would wobble on repeat sampling), and the fixed critical value 1.6449 (how many standard errors on each side capture the middle 90% of a normal distribution). Multiply the standard error by 1.6449 to get the margin, then add and subtract that margin from x-bar to get the upper and lower bounds.
- Enter your sample's average into Sample mean (x-bar) — the point estimate the interval is built around.
- Enter the sample's spread into Sample standard deviation (s) — it cannot be negative.
- Enter how many observations went into that mean into Sample size (n) — larger n narrows the interval.
- Read Lower bound (90% CI) and Upper bound (90% CI) — the true population mean is estimated to lie in that range with 90% confidence.
- Need a wider, more cautious range for a higher-stakes decision? Use this site's 95% or 99% confidence interval instruments instead — same formula, larger critical value.
Worked example — sample mean 100, s = 15, n = 25
Enter 100 into Sample mean (x-bar), 15 into Sample standard deviation (s), and 25 into Sample size (n). The instrument first finds the standard error: 15 / √25 = 15 / 5 = 3.0000. It then multiplies that by the fixed 90% critical value: 1.6449 × 3.0000 = 4.9347 — the margin of error.
Lower bound (90% CI) reads 95.0653 (100 − 4.9347) and Upper bound (90% CI) reads 104.9347 (100 + 4.9347). Reported in full: you are 90% confident the true population mean lies between 95.0653 and 104.9347, a range of just under 9.87 points wide around the sample mean of 100.
Questions
Why is a 90% interval narrower than a 95% or 99% interval?
Because the critical value shrinks as the confidence level drops: 1.6449 for 90%, versus 1.96 for 95% and 2.5758 for 99%, all multiplying the same standard error. A narrower interval is more precise-looking, but it also has a higher chance — 10%, versus 5% or 1% — of failing to capture the true population mean. You are trading certainty for precision, not getting a 'better' answer.
When is a 90% confidence interval the right choice?
When a tighter range is more useful than the extra certainty a wider one buys — early-stage A/B test reads, internal business dashboards, or exploratory quality-control checks where a rough signal matters more than airtight confidence. For published research, clinical claims, or any decision where being wrong is costly, 95% or 99% is the more conventional and defensible choice.
Where does the critical value 1.6449 come from?
It is the z-value on the standard normal distribution beyond which exactly 5% of the area lies in each tail, leaving 90% in the middle. This is the same normal critical value published in standard statistical tables, including the NIST/SEMATECH e-Handbook's table of the standard normal distribution, and it only applies when the standard normal (not t) distribution is the appropriate model — typically a reasonably large sample.
Does this instrument use the t-distribution instead of the normal distribution?
No — it uses the fixed normal critical value 1.6449 throughout, which is the standard approach when the sample is reasonably large or the population standard deviation is treated as known. For small samples where s is only an estimate of an unknown population sigma, a t-distribution critical value (which depends on degrees of freedom and is always somewhat larger) is the more rigorous choice — see this site's t-statistic instrument.
What does '90% confidence' actually mean about my specific interval?
It describes the reliability of the method, not a probability about this one result. Either the true mean is inside your computed interval or it isn't — there's no in-between for a specific interval. What '90%' promises is that if you repeated the sampling and interval-building process many times, about 90% of the resulting intervals would contain the true mean.
Can I use this for any kind of data, or does the sample need to be normal?
The normal critical value 1.6449 is exact when the underlying population is normal, and a very good approximation for other distributions once your sample size is reasonably large (a rule of thumb is n ≥ 30), thanks to the central limit theorem. For small samples from a non-normal population, treat the resulting interval as an approximation rather than an exact result.