Build a z-based confidence interval for a mean from the sample mean, standard deviation, and sample size.
| Lower | – |
| Upper | – |
| Margin of error | – |
Use this calculator for a large-sample (or known-σ) interval around a mean: mean ± z × s / √n. Polls, lab means, and process averages often quote a 95% interval this way when n is large.
The margin of error is z × s / √n. Lower = mean − MoE and upper = mean + MoE. Common z values are 1.645 (90%), 1.96 (95%), and 2.576 (99%). A mean of 50, s = 10, n = 100, and z = 1.96 gives about 48.0 to 52.0.
This uses a normal critical value, not Student's t. For small n with estimated s, a t critical value is more appropriate. The interval is for a mean, not for a single new observation, and it assumes independent draws with the stated spread.
Mean: Sample mean.
Standard deviation (s): Sample (or known) standard deviation; must be positive.
Sample size (n): Number of observations; must be positive.
z: Normal critical value. 1.96 is the usual 95% choice.
Lower / Upper: Endpoints of the interval.
Margin of error: Half-width, z × s / √n.
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