Confidence Interval Calculator
Build a confidence interval for a mean from your sample mean, standard deviation and size — with the margin of error and the interval bounds.
How it works
CI = mean ± z × (sd ÷ √n), with z = 1.645, 1.96 or 2.576 for 90%, 95% or 99%
A confidence interval brackets the range of plausible values for the true population mean. The standard error sd ÷ √n measures how much the sample mean would wobble from sample to sample; multiplying it by the z-value for your confidence level gives the margin of error, which is added and subtracted from the sample mean. This calculator uses the normal (z) approximation, appropriate when the sample is large; for small samples the t-distribution gives a slightly wider, more honest interval.
Worked example
A sample of 30 with mean 100 and standard deviation 15 at 95% confidence: the standard error is 15 ÷ √30 ≈ 2.7386, so the margin of error is 1.96 × 2.7386 ≈ 5.37. The interval is 100 ± 5.37, or about 94.63 to 105.37.
Frequently asked questions
What does "95% confident" actually mean?
It describes 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. It is not a 95% chance that this particular interval does.
Why does a bigger sample give a narrower interval?
The margin of error shrinks with √n. Quadrupling the sample size halves the standard error and therefore halves the width of the interval — precision improves, but with diminishing returns.
When should I use a t-interval instead?
When the sample is small (a common rule of thumb is under 30) and the population standard deviation is unknown. The t-distribution has heavier tails, producing a wider interval that accounts for the extra uncertainty of estimating the spread.
Related calculators
For education — check assumptions (sample size, distribution) before using statistics in real decisions.