Confidence Interval for a Mean Calculator
Build a z-based confidence interval for a population mean from your sample mean, standard deviation and size — with the margin of error and bounds.
How it works
CI = x̄ ± z · (s ÷ √n) · where z is the two-sided critical value for the confidence level
A confidence interval for a mean surrounds your sample average with a margin of error, giving a plausible range for the unknown population mean. The margin is the standard error of the mean (s ÷ √n) scaled by a z multiplier — 1.96 for 95% confidence. Larger samples shrink the standard error, so the interval tightens roughly with the square root of n. This z form assumes a reasonably large sample or a known population spread; for small samples a t-multiplier is more precise.
Worked example
A sample of 36 items has mean 100 and standard deviation 15. The standard error is 15 ÷ √36 = 2.5, and at 95% confidence the margin is 1.96 · 2.5 = 4.9. The interval is 100 ± 4.9, or 95.1 to 104.9.
Sources: Confidence interval — Wikipedia
Frequently asked questions
What does "95% confident" actually mean?
It refers to the method, not one interval: if you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true mean. Any single interval either contains it or does not.
Should I use z or t?
z is fine when the sample is large (roughly n ≥ 30) or the population standard deviation is known. For small samples from a normal population, the t-distribution with n − 1 degrees of freedom gives slightly wider, more accurate intervals.
How can I make the interval narrower?
Collect a larger sample (the margin falls with √n), reduce measurement variability, or accept a lower confidence level. Going from 99% to 90% confidence, for instance, uses a smaller multiplier and tightens the range.
Related calculators
For education — check the assumptions (sample size, independence, distribution) before using these results in real decisions.