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Statistics — Estimation
Confidence Interval Calculator
Enter your sample mean, standard deviation, size and confidence level to get the confidence interval for the mean, with its margin of error.
Confidence interval
—
Range likely to contain the true mean.
| Margin of error ± | — |
|---|---|
| Lower bound | — |
| Upper bound | — |
| Standard error s ÷ √n | — |
| Critical value z | — |
Show our math
Enter your figures above to see the step-by-step working.
Uses the z approximation — a t-interval is better for small samplesEverything computed client-side — nothing leaves this page
How the Confidence Interval Calculator Works
A confidence interval gives a range of plausible values for a population mean, based on a sample. A 95% interval means that if you repeated the sampling many times, about 95% of the intervals built this way would contain the true mean. Enter your sample mean, standard deviation, size and confidence level to get the interval.
This calculator uses the normal (z) approximation, which is appropriate when the sample is reasonably large (roughly n of 30 or more) or the population standard deviation is known. For a small sample from a normal population, a t-interval — which uses a slightly wider critical value — is more accurate.
CI = x̄ ± z · ( s ÷ √n )
Frequently Asked Questions
What does a 95% confidence interval actually mean?
It means the method used to build the interval captures the true population mean about 95% of the time over repeated sampling. It does not mean there is a 95% probability the true mean lies in this one particular interval — the true mean is fixed, the interval is what varies.
Should I use a z or a t interval?
Use z when the sample is large (about 30 or more) or the population standard deviation is known. Use a t interval for a small sample from a normal population; it is slightly wider. This tool uses the z approximation.
How do I make the interval narrower?
Increase the sample size (the interval shrinks with the square root of n), accept a lower confidence level, or reduce variability in what you are measuring. Larger samples give more precise estimates.
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