Standard error of the mean and interpretation
Standard error uses the original measurement unit and describes estimated variability of an average across samples. The estimated variance of the mean uses the squared unit.
To calculate sample or population standard deviation from raw observations, use the Standard Deviation Calculator and follow its own input conventions.
The formula
Divide the supplied sample standard deviation by the square root of sample size. This assumes independent observations with a common variance; the page does not infer an effective size for correlated measurements.
Verified worked example
A sample standard deviation of 10 with 100 independent observations gives standard error 10 / √100 = 1. The estimated variance of the mean is 1.
How to use this calculator
- Enter the sample standard deviation, rather than the variance.
- Enter the whole number of independent observations, at least two.
- Read the mean’s standard error and check whether the sampling assumptions apply.
Standard Error input conventions
Standard deviation describes spread among individual observations. Standard error describes estimated sampling variability of their average. These quantities have the same measurement unit, but answer different questions. Enter a standard deviation already calculated using the appropriate sample convention; this page does not derive it from raw observations.
Interpreting standard error of the mean
The square-root relation assumes independent observations with a common finite variance. Repeated readings from the same person, clustered survey samples, and autocorrelated time series can have a different effective sample size. Increasing the entered count without accounting for dependence can understate uncertainty. The page does not apply a finite-population correction or turn the result into a confidence interval, which would require a critical value and additional assumptions.
Assumptions & limitations
What this calculation assumes
- Observations are independent with a common finite variance, and the supplied deviation is a sample standard deviation.
What to keep in mind
- Mean standard error only; no clustered-sample adjustment, survey weights, or finite-population correction.
Common questions
Does four times the sample size halve standard error?
Under the same variance and independent-sampling assumptions, yes. The denominator changes by the square root of four, which is two.
Can a standard error be zero?
A zero supplied standard deviation produces zero here. It does not prove that a measurement process or a target population has no uncertainty.
Should I enter variance instead of standard deviation?
No. If you have variance, take its nonnegative square root first. Entering variance directly would give the wrong scale and units.
Sources & further reading
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