Read a confidence interval for a mean

From a sample mean, spread, size, and confidence level to the interval — z or t critical value, margin of error, and what the range means.

Tools used in this guide

Confidence interval

The job

A sample mean is a point estimate; the confidence interval says where the true mean plausibly sits. The width depends on the spread, the sample size, and how confident you want to be.

Compute the interval

Enter the sample mean, standard deviation (or known σ), sample size, and confidence level. Choose z for a known population σ or t for a sample SD. The result is the critical value, the margin of error, and the interval bounds.

Use t when σ is estimated from the sample — it widens the interval correctly at small n.
95% is conventional; 90% narrows the interval, 99% widens it.

Worked example

Mean 50, sample SD 8, n = 40, 95% with t: critical ≈ 2.023, margin ≈ 2.56, interval 47.44 to 52.56.

Where it stops

Normal or t-based intervals for a mean only — no proportions, no paired designs, no bootstrap. Sample size is capped at 1e9: the z interval needs n ≥ 2, the t interval n ≥ 3. Confidence levels are the three presets — 90%, 95%, 99%. Interpretation is approximate under non-normality, and the t critical value is an approximation that improves with larger samples.