Sxxの式展開

$S_{xx}$の式展開

$$\begin{array}{rcl}{S}_{xx}& =& \sum _{i=1}^n{(x_i-\overline{x})}^2\\ & =& \sum _{i=1}^n\{x_i^2-2\overline{x}{x}_i+{\overline{x}}^2\}\\ & =& \sum _{i=1}^n{x}_i^2-2\overline{x}\sum _{i=1}^n{x}_i+{\overline{x}}^2\sum _{i=1}^{n}1\\ & =& \sum _{i=1}^n{x}_i^2-2\overline{x}n\overline{x}+{\overline{x}}^{2}n\\ & =& \sum _{i=1}^n{x}_i^2-2n{\overline{x}}^2+n{\overline{x}}^2\\ & =& \sum _{i=1}^n{x}_i^2-n{\overline{x}}^2\end{array}$$