Standard Error Calculator
How precisely your sample mean estimates the true mean — the standard error, with the confidence interval it produces.
The standard error of the mean is σ ÷ √n.
How the standard error calculator works
The standard error of the mean is σ ÷ √n. It is not the spread of the data — that is the standard deviation — but the spread of the *sample mean* across hypothetical repeated samples.
The √n is the whole story of sampling: to halve the standard error you must quadruple the sample. That is why survey precision gets expensive so quickly.
Formula: SE = s / √n; CI = x̄ ± t × SE
Worked examples
| Inputs | Standard error of the mean | Note |
|---|---|---|
| Eight measurements | 0.626783 | SE ≈ 0.62 |
| SD 10, n 100 | 1 | SE = 1 |
| SD 10, n 400 | 0.5 | SE = 0.5 — four times the sample, half the error |
FAQFrequently asked questions
What is the difference between SD and SE?
The standard deviation describes the spread of the data; the standard error describes the precision of the mean. The SE is always the smaller of the two.
Why divide by √n?
Because averaging cancels independent errors, and the cancellation goes as the square root of the count.
How do I halve my standard error?
Quadruple the sample. There is no cheaper route — this is the fundamental economics of sampling.
Should error bars show SD or SE?
Say which. SE bars look reassuringly small and are often used to make a difference look significant; SD bars show what the data actually did.
Why a t-value rather than 1.96?
Because the standard deviation is estimated from the same small sample. The t distribution widens the interval to account for that, converging on 1.96 as n grows.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.