Sxyの式展開

\(Sxy\)の式展開

$$ \begin{eqnarray} S_{xy}&=&\sum_{i=1}^n\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right) \\&=&\sum_{i=1}^n\left\{ \left(x_i-\bar{x}\right)y_i -\left(x_i-\bar{x}\right)\bar{y} \right\} \\&=&\sum_{i=1}^n\left(x_i-\bar{x}\right)y_i-\bar{y}\sum_{i=1}^n\left(x_i-\bar{x}\right) \\&=&\sum_{i=1}^n\left(x_i-\bar{x}\right)y_i-\bar{y}\left(\sum_{i=1}^nx_i-\bar{x}\sum_{i=1}^n1\right) \\&=&\sum_{i=1}^n\left(x_i-\bar{x}\right)y_i-\bar{y}\left(n\bar{x}-\bar{x}n\right) \;\cdots\;\bar{x}=\frac{1}{n}\sum_{i=1}^nx_i,\;n\bar{x}=\sum_{i=1}^nx_i \\&=&\sum_{i=1}^n\left(x_i-\bar{x}\right)y_i-\bar{y}\cdot0 \\&=&\sum_{i=1}^n\left(x_i-\bar{x}\right)y_i \\&=&\sum_{i=1}^nx_iy_i-\bar{x}\sum_{i=1}^ny_i \\&=&\sum_{i=1}^nx_iy_i-\bar{x}n\bar{y} \\&=&\sum_{i=1}^nx_iy_i-n\bar{x}\bar{y} \end{eqnarray} $$