How this calculation works
For independent observations from a population with finite variance, larger samples reduce the variability of their means. An estimated standard deviation produces an estimated standard error.
Standard error = standard deviation ÷ √sample sizeA worked example
A standard deviation of 12 with 36 observations gives a standard error of 2.
Good to know
Assumes the idealized model described above. Use consistent units and independently verify figures before a consequential decision.
Four times the sample size halves the standard error
The square root in the denominator makes sample-size gains gradual. With a fixed standard deviation, increasing n from 25 to 100 halves the standard error. Doubling n reduces it by a factor of √2 rather than by a factor of two.
This is why squeezing a much smaller sampling margin from an already large study can require many additional observations. More data help the model, but the improvement is not proportional to the raw count.
Spread of observations and uncertainty of the mean differ
Standard deviation describes variability among individual observations. Standard error describes variability of a sample mean across repeated samples under the assumptions. A small standard error does not imply every individual value is close to the mean.
The simple formula assumes an appropriate independent-sampling model. Clustering, serial correlation or biased selection can invalidate the apparent precision; a large n does not automatically correct those problems.
A few questions, answered
Can I use a sample standard deviation?
Yes, as an estimate. The resulting standard error is also estimated.
Does this report a confidence interval?
No. An interval also needs an appropriate distributional model and critical value.