Logan Nielsen
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Derivation · Quantitative Finance / Econometrics

Derivation of the Variance of the Sample Mean

A compact derivation of the variance of the sample mean.

August 12, 2021 · 2 min read

By definition, Xˉ=(X1+X2+⋯+Xn)/n\bar X = (X_1 + X_2 + \dots + X_n)/n.

Therefore:

Var(Xˉ)=Var(X1/n+X2/n+⋯+Xn/n)=(1/n2)Var(X1+X2+⋯+Xn)/n2=σ2/n\begin{align} Var(\bar X) &= Var(X_1/n + X_2/n + \dots + X_n/n) \\ &= (1/n^2) Var(X_1 + X_2 + \dots + X_n)/n^2 \\ &= \sigma^2/n \end{align}