Standard Error of the Mean Calculator
SEM = s / √n — how precisely your sample mean estimates the true population mean. Plus a 90/95/99% confidence interval (z or Student's t).
SEM & confidence interval from summary stats
SEM from raw data (paste values)
Paste numbers separated by commas, spaces, or new lines. Uses the sample SD (÷ n−1), then SEM = s/√n.
t-critical values (two-sided)
Computed live by this page's own inverse-t routine (same code as the calculator). t* for 90% uses the 0.95 quantile, 95% the 0.975 quantile, 99% the 0.995 quantile.
| df | t* 90% | t* 95% | t* 99% |
|---|
SEM vs standard deviation — not the same thing
- SD (s) describes the spread of individual data points around the mean.
- SEM (s/√n) describes the precision of the sample mean itself as an estimate of the population mean.
- SEM always shrinks as n grows — even if the SD stays the same. Quadrupling n halves the SEM.
- A 95% CI ≈ x̄ ± 2×SEM is a quick mental check of where the true mean plausibly lies.
Scope boundary
This tool answers one-sample questions: "how precise is my sample mean?" (SEM = s/√n) and "what is the confidence interval for the mean?". It does not do:
- Multi-group pooled SD — combining variances across 2–5 groups is the job of the pooled standard deviation tool (#1349), not this one.
- Descriptive mean/SD of a dataset — the legacy average and standard deviation calculator computes the plain SD; this tool goes one step further into inferential statistics (SEM, CI).
Worked examples & FAQ
The standard error of the mean (SEM) is the standard deviation of the sampling distribution of the mean: SEM = s / √n, where s is the sample standard deviation and n is the sample size. It tells you how far your sample mean is likely to be from the true population mean.
SD describes how spread out the individual observations are. SEM describes how precisely the sample mean estimates the population mean. Two samples can have the same SD but very different SEMs if their sample sizes differ.
SEM = 10 / √100 = 10 / 10 = 1.0000. With a sample of 100, the sample mean is typically within about 1 unit of the true mean.
Because SEM = s/√n, doubling n divides the SEM by √2 ≈ 1.4142. Example with s = 8: at n = 25 the SEM is 1.6000; at n = 100 (4× the sample) it is 0.8000 — exactly half, since √4 = 2.
SEM = 10/√30 = 1.8257. With df = 29, t* = 2.0452. Margin = 2.0452 × 1.8257 = 3.7341. 95% CI = 50 ± 3.7341 = (46.2659, 53.7341).
Mean = 5.0000, sample SD = 2.1381, n = 8. SEM = 2.1381/√8 = 0.7559. With df = 7, t* = 2.3646, the 95% CI for the mean is (3.2125, 6.7875).
Use Student's t (df = n − 1) whenever the population SD is unknown — which is almost always. For large n the t critical value converges to z (for df = 29, t* = 2.0452 vs z = 1.9600). This calculator offers both so you can compare.
SEM itself is still defined (s/√1 = s), but a t-based confidence interval needs df = n − 1 ≥ 1, i.e. n ≥ 2. This calculator reports an honest error instead of a meaningless interval.
Free tool by NRTechStudio