Sxxの式展開

\(S_{xx}\)の式展開

$$ \begin{eqnarray} S_{xx}&=&\sum_{i=1}^n\left(x_i-\bar{x}\right)^2 \\&=&\sum_{i=1}^n\left\{x_i^2-2\bar{x}x_i+\bar{x}^2\right\} \\&=&\sum_{i=1}^nx_i^2-2\bar{x}\sum_{i=1}^nx_i+\bar{x}^2\sum_{i=1}^n1 \\&=&\sum_{i=1}^nx_i^2-2\bar{x}n\bar{x}+\bar{x}^2n \\&=&\sum_{i=1}^nx_i^2-2n\bar{x}^2+n\bar{x}^2 \\&=&\sum_{i=1}^nx_i^2-n\bar{x}^2 \end{eqnarray} $$