Confidence Interval Calculator
Build a 90%, 95%, or 99% confidence interval for a mean from the sample mean, standard deviation, and size.
95% confidence interval
46.08 to 53.92
x̄ ± z × (s / √n) = 50 ± 3.92
This is a z-interval, which assumes a known or large-sample standard deviation. For small samples you'd normally use a t-interval, whose critical value is a little wider.
How it works
A confidence interval is a range around your sample mean that's likely to contain the true population mean. A 95% interval, for instance, is built so that if you repeated the study many times, about 95% of the intervals would capture the real value.
The formula is x̄ ± z × (s / √n): take the sample mean, then add and subtract a margin of error made of the critical z-score times the standard error. This tool uses z = 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%. You can type the mean, standard deviation, and sample size directly, or paste a data set and let it work them out.
Notice how the interval tightens as the sample grows: the √n in the denominator shrinks the standard error, so bigger samples give more precise estimates. A higher confidence level, on the other hand, widens the interval because you're demanding more certainty.
Frequently asked questions
What do the 90%, 95%, and 99% levels mean?
They're the long-run capture rate of the method. At 95%, if you drew many samples and built an interval from each, roughly 95% of those intervals would contain the true mean. Higher confidence uses a larger z-score, which makes the interval wider.
Should I use a z-interval or a t-interval?
This calculator uses z-scores, which suit large samples or a known population standard deviation. For small samples with an estimated standard deviation, a t-interval is more accurate — its critical value is a bit larger, giving a slightly wider interval.
Can I paste raw data instead of summary stats?
Yes. Paste two or more numbers into the data box and the tool computes the mean, the sample standard deviation, and the count for you, then builds the interval from those.