The standard error of the mean measures how much a sample mean would vary if you drew the sample again. Paste your data below and it is calculated alongside the standard deviation it derives from.
Separate values with commas, spaces, tabs or new lines — paste a column straight from a spreadsheet and it will parse. Decimals and negatives are fine.
The wide, faint curve is the spread of the individual values (s = 3.044). The narrow curve is the spread of the sample mean (SE = 1.076) — √8 times tighter, because averaging cancels noise.
Show the working, step by step
The formula
SE = s / √n
where s is the sample standard deviation and n the sample size.
That is the whole calculation — the difficulty is not arithmetic but knowing which of SD and
SE answers the question in front of you.
Standard deviation or standard error?
| Standard deviation | Standard error | |
|---|---|---|
| Describes | Spread of the observations | Precision of the mean |
| Answers | "How different are individuals?" | "How well do I know the average?" |
| As n grows | Settles on the population's true spread | Shrinks towards zero |
| Formula | √[Σ(x − x̄)²/(n − 1)] | s / √n |
| Use it for | Describing your data | Confidence intervals, significance tests |
A concrete case. Adult heights have a standard deviation of roughly 7 cm, and that is a fact about people — measuring a million of them will not change it. But the standard error of the mean height from a sample of 100 is 7/√100 = 0.7 cm, and from a sample of 10,000 it is 0.07 cm. More data does not make people more alike; it makes your estimate of their average sharper.
How sample size drives it
| n | √n | SE when s = 10 |
|---|---|---|
| 5 | 2.24 | 4.47 |
| 25 | 5.00 | 2.00 |
| 100 | 10.00 | 1.00 |
| 400 | 20.00 | 0.50 |
| 1600 | 40.00 | 0.25 |
Each halving of the standard error costs four times the data. This diminishing return is the central constraint in study design: it is why sample sizes are chosen against a target precision rather than picked for roundness, and why very precise estimates are so expensive.
Recover a standard deviation from a published standard error
Papers frequently report "mean ± SE" without giving the standard deviation, which is a problem if you want to pool their result with others. Multiplying back by √n recovers it exactly — no approximation is involved, because SE = s/√n is a definition.
The wide, faint curve is the spread of the individual values (s = 4.217). The narrow curve is the spread of the sample mean (SE = 0.77) — √30 times tighter, because averaging cancels noise.
Show the working, step by step
If the paper reported a confidence interval instead of a standard error, the standard deviation can still be recovered — see the confidence interval page for the relationship, or use the Cochrane Handbook's method of dividing the interval width by 2t*.
Related calculators
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Standard deviation calculator
The spread of the data itself.
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Confidence interval
Turns a standard error into a range for the mean.
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Variance calculator
The squared spread underneath both.
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Z-score calculator
Standardising a single value rather than a mean.
Common questions
What is the difference between standard deviation and standard error?
The standard deviation describes how spread out your data is. The standard error describes how much your sample mean would move around if you repeated the study. They answer different questions and are not interchangeable.
The standard error is always the smaller of the two, by a factor of √n, and it keeps shrinking as you collect more data. The standard deviation does not — it converges on the real spread of the population, which more data cannot change.
Should error bars show SD or SE?
Show the standard deviation if you want the reader to see how variable the individual observations are. Show the standard error or a confidence interval if you want them to judge how precisely the mean is known.
SE bars are always narrower, which is why they are sometimes chosen for the wrong reason. Whichever you use, state it in the caption — a bar with no label is uninterpretable.
Why does the standard error shrink with a larger sample?
Because averaging cancels noise. Individual values scatter by σ no matter how many you collect, but their average becomes more stable as the high and low draws increasingly offset one another. The rate of that stabilisation is exactly 1/√n.
The square root is the reason large studies get expensive: halving the standard error requires four times the data, and halving it again requires sixteen times.
Can I get the standard deviation back from a published standard error?
Yes, exactly: s = SE × √n. That recovery is used routinely in meta-analysis
to reconstruct SDs from papers that only reported standard errors. The second calculator on
this page does it.