Confidence Interval for the Mean
Compute a normal-CI around the mean of a numeric sample.
Overview
The Confidence Interval for the Mean tool builds a symmetric range around your sample mean that, with a chosen confidence level (usually 95%), is expected to contain the true population mean. Paste a list of numbers, pick a level and read off the lower and upper bounds.
It is built for researchers reporting summary statistics, product analysts adding error bars to dashboards, students checking textbook answers and engineers gauging measurement precision. Without an interval, a mean is just one number — with one, it becomes a statement about plausible values.
How it works
Given sample size n, sample mean x̄ and sample standard deviation s, the standard error is SE = s / sqrt(n). The confidence interval is x̄ ± t * SE, where t is the critical value from the t-distribution with n - 1 degrees of freedom at the chosen confidence level.
For 95% confidence and large n the t-value approaches 1.96 — the familiar normal-distribution z-score. For small samples, t is larger, widening the interval to account for uncertainty about the population variance.
Examples
Sample [10, 12, 14, 11, 13], 95% CI
→ mean 12.0, CI ≈ [10.04, 13.96]
Sample [98, 100, 102, 99, 101, 100], 99% CI
→ mean 100.0, CI ≈ [97.78, 102.22]
30 measurements, mean 50, std 5, 95% CI
→ CI ≈ [48.13, 51.87]
FAQ
What does 95% confidence actually mean?
If you repeated the experiment many times and built the same interval each time, about 95% of those intervals would contain the true mean. It does not mean there is a 95% probability the true mean lies in this single interval.
Why is the interval wider for small samples?
With less data, your estimate of the population standard deviation is less reliable, so the t-distribution has fatter tails and a larger critical value.
Do I need normality?
For small samples the t-CI assumes the data are roughly normal. For large samples (n > 30) the central limit theorem makes the interval robust to non-normality.
What's the difference between confidence and prediction intervals?
A confidence interval bounds the mean; a prediction interval bounds where a single new observation might land. Prediction intervals are always wider.
Can I get a one-sided interval?
This tool computes two-sided intervals. For a one-sided bound at confidence c, recompute with confidence 2c - 1 and take only the relevant tail.