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Standard error of an independent mean

For independent, identically distributed observations with finite variance, the mean has standard deviation sigma divided by square root of n.

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Track the variance of the average

The average adds n observations and divides by n. Independence makes the variance of the sum n times the common variance. Division by n squares the divisor in the variance calculation, leaving sigma²/n. Taking its square root gives sigma/√n.

Var⁡(Xˉ)=σ2n,SE=σn\operatorname{Var}(\bar X)=\frac{\sigma^2}{n},\quad SE=\frac{\sigma}{\sqrt n}

This relationship requires independent observations with the same finite variance.

Read two different scales

Fictional scale comparison
Quantityn = 25, sigma = 4n = 100, sigma = 4
Individual-observation SD44
Mean SE0.80.4

Label an estimated standard error

Replacing unknown sigma with a sample standard deviation s estimates the uncertainty. It does not make sigma known. In a normal one-sample setting, that distinction leads to a t reference for a standardized mean.

Inspect dependence before using the shortcut

For observations that move together, covariance terms enter the variance of their sum. Merely counting more correlated return observations does not prove that uncertainty shrinks at the independent-sample rate. The data design determines whether this formula is appropriate.

Further references