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英語版ページに↓みたいなこと書いてあるよ


This statement is called the Bienayme' formula.[1] and was discovered in 1853.
It is often made with the stronger condition that the variables are independent,
but uncorrelatedness suffices. So if the variables have the same variance σ2,
then, since division by n is a linear transformation, this formula immediately
implies that the variance of their mean is

\operatorname{Var}(\overline{X}) = \operatorname{Var}\left(\frac{1}{n}\sum_{i=1}^n X_i\right) = \frac {1}{n^2} n \sigma^2 = \frac {\sigma^2} {n}.