標本の二乗和

標本分散の式展開

$$\begin{array}{rcl}{\displaystyle \sum _{k=1}^n(X_k-\bar{X}{)}^2}& =& {\displaystyle \sum _{k=1}^n(X_k^2-2\bar{X}{X}_k+{\bar{X}}^2)}\\ & =& {\displaystyle \sum _{k=1}^n{X}_k^2-2\bar{X}\sum _{k=1}^n{X}_k+{\bar{X}}^2\sum _{k=1}^{n}1}\\ & =& {\displaystyle \sum _{k=1}^n{X}_k^2-2n{\bar{X}}^2+n{\bar{X}}^2}\\ & & {\displaystyle \dots \sum _{k=1}^n{X}_k=n\bar{X}}\\ & =& {\displaystyle \sum _{k=1}^n{X}_k^2-n{\bar{X}}^2}\end{array}$$

標本の二乗和

$$\begin{array}{rcl}{\displaystyle \sum _{k=1}^n{X}_k^2}& =& {\displaystyle \sum _{k=1}^n(X_k-\bar{X}{)}^2+n{\bar{X}}^2}\end{array}$$