標本の二乗和

標本分散の式展開

$$\begin{array}{rcl} \displaystyle \sum_{k=1}^{n}(X_k-\overline{X})^2 &=&\displaystyle \sum_{k=1}^{n}(X_k^2-2\overline{X} X_k+\overline{X}^2)\\ &=&\displaystyle \sum_{k=1}^{n}X_k^2-2\overline{X} \sum_{k=1}^{n}X_k+\overline{X}^2\sum_{k=1}^{n}1\\ &=&\displaystyle \sum_{k=1}^{n}X_k^2-2n\overline{X}^2+n\overline{X}^2\\ &&\displaystyle\,\dotso\,\sum_{k=1}^{n}X_k=n\overline{X}\\ &=&\displaystyle \sum_{k=1}^{n}X_k^2-n\overline{X}^2\\ \end{array}$$

標本の二乗和

$$\begin{array}{rcl} \displaystyle \sum_{k=1}^{n}X_k^2 &=&\displaystyle \sum_{k=1}^{n}(X_k-\overline{X})^2+n\overline{X}^2\\ \end{array}$$