変量の変換

${\displaystyle \overline{y}=\frac{1}{n}\left(y_1+y_2+\cdots +y_n\right)}$

${\displaystyle =\frac{1}{n}\left\{a\left(x_1+x_2+\cdots +x_n\right)+nb\right\}}$

${\displaystyle =a\cdot \frac{1}{n}\left(x_1+x_2+\cdots +x_n\right)+b}$

${\displaystyle =a\overline{x}+b}$

●分散 ${s_y}^2$

${\displaystyle s_y{}^2=\frac{1}{n}\left\{{\left(y_1-\overline{y}\right)}^2+{\left(y_2-\overline{y}\right)}^2+\cdots +{\left(y_n-\overline{y}\right)}^2\right\}}$

分散の式を展開し，整理した結果より

${\displaystyle =\frac{1}{n}\left(y_1{}^2+y_2{}^2+\cdots +y_n{}^2\right)-{\left(\overline{y}\right)}^2}$

${\displaystyle =\frac{1}{n}\left\{{\left(a{x}_1+b\right)}^2+{\left(a{x}_2+b\right)}^2+\cdots +{\left(a{x}_n+b\right)}^2\right\}}$ ${\displaystyle -{\left(a\overline{x}+b\right)}^2}$

${\displaystyle =\frac{1}{n}\left\{\left(a^2{x}_1{}^2+2ab{x}_1+b^2\right)+\left(a^2{x}_2{}^2+2ab{x}_2+b^2\right)+\cdots +\left(a^2{x}_2{}^2+2ab{x}_n+b^2\right)\right\}}$ ${\displaystyle -\left\{a^2{\left(\overline{x}\right)}^2+2ab\overline{x}+b^2\right\}}$

${\displaystyle =\frac{1}{n}\left\{a^2\left(x_1{}^2+x_2{}^2+\cdots +x_n{}^2\right)+2ab\left(x_1+x_2+\cdots +x_n\right)+n{b}^2\right\}}$ ${\displaystyle -a^2{\left(\overline{x}\right)}^2-2ab\overline{x}-b^2}$

${\displaystyle =a^2\cdot \frac{1}{n}\left(x_1{}^2+x_2{}^2+\cdots +x_n{}^2\right)}$ ${\displaystyle +2ab\cdot \frac{1}{n}\left(x_1+x_2+\cdots +x_n\right)}$ ${\displaystyle +b^2-a^2{\left(\overline{x}\right)}^2-2ab\overline{x}-b^2}$

${\displaystyle =a^2\cdot \frac{1}{n}\left(x_1{}^2+x_2{}^2+\cdots +x_n{}^2\right)}$ ${\displaystyle +2ab\overline{x}+b^2-a^2{\left(\overline{x}\right)}^2-2ab\overline{x}-b^2}$

${\displaystyle =a^2\cdot \frac{1}{n}\left(x_1{}^2+x_2{}^2+\cdots +x_n{}^2\right)-a^2{\left(\overline{x}\right)}^2}$

${\displaystyle =a^2\left\{\frac{1}{n}\left(x_1{}^2+x_2{}^2+\cdots +x_n{}^2\right)-{\left(\overline{x}\right)}^2\right\}}$

${\displaystyle =a^2{s}_x{}^2}$

●標準偏差 $s_y$

${\displaystyle s_y=\sqrt{s_y{}^2}=\sqrt{a^2{s}_x{}^2}=\left|a\right|s_x}$