Standard Error of the Mean Calculator
Compute the standard error of the mean (SEM) from a sample standard deviation and size — how precisely your sample average estimates the true mean.
How it works
SEM = s ÷ √n
The standard error of the mean measures how much a sample average would bounce around if you repeated the study — it is the standard deviation of the sampling distribution of the mean. Unlike the sample standard deviation, which describes the spread of individual data points, the SEM describes the precision of the average itself. Because it divides by √n, quadrupling the sample halves the standard error, which is why bigger samples give steadier estimates.
Worked example
A sample of 25 observations has a standard deviation of 20. The standard error is 20 ÷ √25 = 20 ÷ 5 = 4, so the sample mean is precise to about ±4 (before applying a confidence multiplier).
Sources: Standard error — Wikipedia
Frequently asked questions
What is the difference between standard deviation and standard error?
Standard deviation describes how spread out individual values are. Standard error describes how spread out the sample mean would be across repeated samples — it is always smaller, and it shrinks as the sample grows.
Which should I report in a chart?
Show the standard deviation when you want readers to see the variability of the data, and the standard error (or a confidence interval) when you want to convey how precisely you have pinned down the mean. Mislabelling the two is a common source of confusion.
How do I turn the SEM into a confidence interval?
Multiply the SEM by the critical value for your confidence level (about 1.96 for 95%) to get the margin of error, then add and subtract it from the sample mean.
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For education — check the assumptions (sample size, independence, distribution) before using these results in real decisions.