Standard Error of the Mean Calculator (σ ÷ √n)
The standard error of a sample mean from the standard deviation and sample size, with the 95% error bound and how n changes it.
Inputs
- Working
- 8 ÷ √50 = 8 ÷ 7.0711 = 1.1314
- σ
- 8
- n
- 50
- 95% error bound (1.96 × SE)
- ±2.2175
- To halve the standard error
- n = 200
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What each Standard Error of the Mean input means
- Std dev (σ)
- The standard deviation of individual values: σ if known, or the sample standard deviation s.
- Sample size
- The number of observations averaged — at least 2.
Standard Error of the Mean formula
SE = σ ÷ √n · 95% error bound ≈ 1.96 × SE · with a sample standard deviation s, SE ≈ s ÷ √n
Take many samples of size n from the same population and their means will not all be the same; the standard error is the standard deviation of those sample means. The central limit theorem says it equals the population standard deviation divided by the square root of the sample size, and that the sample means are approximately normal whatever the population looks like once n is reasonably large. So the standard error is the unit in which a sample mean's uncertainty is measured: a mean of 90 with a standard error of 3 is telling you the true mean is likely within a few 3s of 90. The square root is the important feature — averaging 4 times as many observations only halves the noise, 100 times as many only cuts it to a tenth.
Standard Error of the Mean worked example
A standard deviation of 8 and a sample of 50: √50 = 7.071, and 8 ÷ 7.071 = 1.131. Sample means of 50 observations therefore scatter with a standard deviation of 1.13 around the true mean, and a 95% error bound is 1.96 × 1.131 = ±2.22. OpenStax's Example 7.1 is the same arithmetic: σ = 15 and n = 25 give 15 ÷ 5 = 3, so X̄ ~ N(μ, 3).
More Standard Error of the Mean examples
OpenStax's Example 7.1: σ = 15 and samples of 25 give 15 ÷ √25 = 15 ÷ 5 = 3, so sample means scatter with a standard deviation of 3, and a 95% bound is ±5.88.
A standard deviation of 12 over 16 observations: 12 ÷ 4 = 3 — the same standard error as above with a smaller spread and a smaller sample. Halving it would take 64 observations.
Where the Standard Error of the Mean calculator can give the wrong answer
- Standard deviation describes the spread of individual values; standard error describes the spread of sample means. Quoting one as the other is the commonest error in charts with error bars — SE bars are always smaller.
- Using the sample standard deviation s in place of σ is standard practice but adds uncertainty for small n, which is why small-sample inference uses t rather than z.
- The formula assumes independent observations. Repeated measurements on the same few subjects, or clustered data, have a larger effective standard error than σ ÷ √n suggests.
- For sampling without replacement from a small population, a finite-population correction √((N − n) ÷ (N − 1)) shrinks the standard error; the page assumes the population is large.
- The standard error of other statistics — a proportion, a difference of means, a regression slope — has its own formula; this page is the mean only.
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Standard Error of the Mean reference table
| Sample size | 1 ÷ √n | Standard error when σ = 10 |
|---|---|---|
| 4 | 0.5 | 5 |
| 9 | 0.333 | 3.33 |
| 16 | 0.25 | 2.5 |
| 25 | 0.2 | 2 |
| 100 | 0.1 | 1 |
| 400 | 0.05 | 0.5 |
| 10,000 | 0.01 | 0.1 |
Standard Error of the Mean FAQ
- Divide the standard deviation by the square root of the number of observations. SD 12 and n = 16: 12 ÷ 4 = 3. If you only have the sample standard deviation, use that; the page accepts either.
- Standard deviation measures how spread out the individual values are. Standard error measures how precisely the sample mean estimates the population mean, and it shrinks as the sample grows while the standard deviation does not. SE = SD ÷ √n.
- Through the square root: n = 100 gives a standard error one-tenth of the standard deviation, n = 400 one-twentieth. To halve the standard error you need four times the observations, which the page shows for whatever n you enter.
- Multiply by z — 1.96 for 95% — to get the error bound, then add and subtract it from the sample mean. The page shows the 95% bound; the confidence interval calculator builds the whole interval.
- It depends on the claim. Standard-deviation bars show how spread out the individual values are; standard-error bars show how precisely each mean is known, and are narrower by a factor of √n. A chart should say which it uses.
- It means a more precise estimate of the mean, which usually comes from more data. It says nothing about whether the mean itself is high or low.