How this instrument works
A confidence interval is a range built around a sample mean that is likely to contain the true population mean. The confidence level — 80%, 90%, 95%, 98% or 99% here — describes the long-run behaviour of the method: at 95% confidence, for instance, about 95% of intervals built this way across repeated sampling would capture the true mean. This instrument does not fix that level; it lets you choose it from a dropdown, applying the matching normal critical value automatically.
Each confidence level corresponds to a specific critical value on the standard normal distribution: 1.2816 for 80%, 1.6449 for 90%, 1.96 for 95%, 2.3263 for 98%, and 2.5758 for 99%. Higher confidence always means a larger critical value and therefore a wider interval on the same data — there is no way to get more certainty and more precision at once from a fixed sample. Selecting the level here is choosing that trade-off directly, rather than being locked into one field convention.
The interval itself is built from three ingredients regardless of which level you pick: the sample mean x-bar (your point estimate), the standard error s/√n (how much that estimate would wobble on repeat sampling), and the chosen critical value z. Multiply the standard error by z 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.
- Pick a level from Confidence level — 80%, 90%, 95%, 98% or 99%, each with its critical value shown alongside it.
- Read Lower bound and Upper bound, and switch the Confidence level dropdown to instantly compare how the same data looks at a different standard.
- Need a fixed, always-95% (or always-90%, always-99%) instrument instead? This site's dedicated 90%, 95% and 99% confidence interval calculators skip the selector entirely.
Worked example — sample mean 100, s = 15, n = 25, at 95%
Enter 100 into Sample mean (x-bar), 15 into Sample standard deviation (s), and 25 into Sample size (n), then select '95% (z = 1.96)' from Confidence level. The instrument first finds the standard error: 15 / √25 = 15 / 5 = 3.0000. It then multiplies that by the selected critical value: 1.96 × 3.0000 = 5.88 — the margin of error.
Lower bound reads 94.12 (100 − 5.88) and Upper bound reads 105.88 (100 + 5.88). Now switch Confidence level to '80% (z = 1.2816)' without touching the other three fields: the margin shrinks to 1.2816 × 3.0000 = 3.84, and the bounds tighten to roughly 96.16 and 103.84 — the same underlying data, a visibly narrower claim.
Questions
How do I choose which confidence level to use?
95% is the field-wide default for most published research and general-purpose reporting. Use 90% or 80% when a tighter, more actionable range matters more than airtight certainty — early-stage analysis or internal screening. Use 98% or 99% for higher-stakes claims, such as safety-critical or regulatory reporting, where the cost of missing the true mean is high.
Why does the interval get wider when I pick a higher confidence level?
Because the critical value z grows as confidence rises — from 1.2816 at 80% up to 2.5758 at 99% — and that value directly scales the margin added and subtracted from the mean. Squeezing more certainty out of the same sample always means casting a wider net; there is no way to raise confidence and shrink the interval simultaneously without collecting more data.
What does the confidence level actually mean about my specific interval?
It describes the reliability of the method, not a probability about this one result. At 95% confidence, for example, if you repeated the sampling and interval-building process many times, about 95% of the resulting intervals would contain the true population mean. Any single computed interval either contains the true mean or it doesn't — there's no partial credit.
Where do the five critical values (1.2816 through 2.5758) come from?
Each is the point on the standard normal distribution beyond which the matching tail probability lies — for example, 1.96 leaves exactly 2.5% in each tail, totalling 5%, for 95% confidence in the middle. These are the standard published two-sided normal critical values found in statistical tables and in the NIST/SEMATECH e-Handbook's table of the standard normal distribution.
Should I use the normal critical values here, or a t-distribution instead?
This instrument's z values are appropriate when the sample is reasonably large (a common rule of thumb is n ≥ 30) or the population standard deviation is treated as known. For smaller samples where s only estimates an unknown population sigma, a t-distribution critical value — always somewhat larger, and dependent on degrees of freedom — is the more rigorous choice; see this site's t-statistic instrument.
How is this different from the site's separate 90%, 95% and 99% calculators?
Those three instruments fix a single standard level so you can jump straight to it without a selector step. This instrument covers the same math across five levels (80%, 90%, 95%, 98%, 99%) in one place, which is useful for directly comparing how the same sample data behaves at different confidence standards side by side.