Sxyの式展開

$Sxy$の式展開

$$\begin{array}{rcl}{S}_{xy}& =& \sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})\\ & =& \sum _{i=1}^n\{(x_i-\overline{x})y_i-(x_i-\overline{x})\overline{y}\}\\ & =& \sum _{i=1}^n(x_i-\overline{x})y_i-\overline{y}\sum _{i=1}^n(x_i-\overline{x})\\ & =& \sum _{i=1}^n(x_i-\overline{x})y_i-\overline{y}(\sum _{i=1}^n{x}_i-\overline{x}\sum _{i=1}^{n}1)\\ & =& \sum _{i=1}^n(x_i-\overline{x})y_i-\overline{y}(n\overline{x}-\overline{x}n)\cdots \overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i,n\overline{x}=\sum _{i=1}^n{x}_i\\ & =& \sum _{i=1}^n(x_i-\overline{x})y_i-\overline{y}\cdot 0\\ & =& \sum _{i=1}^n(x_i-\overline{x})y_i\\ & =& \sum _{i=1}^n{x}_i{y}_i-\overline{x}\sum _{i=1}^n{y}_i\\ & =& \sum _{i=1}^n{x}_i{y}_i-\overline{x}n\overline{y}\\ & =& \sum _{i=1}^n{x}_i{y}_i-n\overline{x}\overline{y}\end{array}$$