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Confidence Interval Calculator

Calculate a confidence interval for a population mean from a sample's mean, standard deviation, and size, at any confidence level — not just the usual 90/95/99% presets.

Critical z-value
1.96
Margin of error
± 4.8999
Confidence interval
95.1001 to 104.8999

How it works

A confidence interval estimates a range likely to contain the true population mean, built as the sample mean plus or minus a margin of error: mean ± z × (standard deviation ÷ √n). The z-value is the standard normal distribution's critical value for your chosen confidence level — for a 95% interval, that's the value leaving 2.5% of the distribution in each tail (≈1.96) — computed here from a standard rational-polynomial approximation of the normal quantile function, so any confidence level works, not just the handful a printed z-table lists.

This is the large-sample interval: it assumes the sampling distribution of the mean is approximately normal (reasonable once n is reasonably large, per the Central Limit Theorem) and treats the sample's own standard deviation as a good stand-in for the population's. Shrinking the margin of error either means a smaller confidence level (a less certain claim) or a larger sample — the margin shrinks with 1/√n, so quadrupling the sample size only halves it.

FAQ

What does "95% confidence" actually mean?

It does not mean there's a 95% chance the true mean falls in this specific interval. It means that if you repeated this sampling process many times and built an interval the same way each time, about 95% of those intervals would contain the true population mean. The true mean is fixed; it's the interval that varies from sample to sample.

Should I use this or a t-distribution interval for a small sample?

This calculator uses the normal (z) distribution, which is the standard choice once the sample is reasonably large (commonly, n of 30 or more). For a small sample — especially if the underlying population isn't known to be roughly normal — a Student's t-interval, which uses a wider critical value that accounts for the extra uncertainty in estimating the standard deviation itself, is the more rigorous choice.

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