変量の変換

変量 $3$X$ の $3$n$ 個のデータ $3$\hspace{1}{x}_{1}$ ， $3$\hspace{1}{x}_{2}$ ， $3$\cdots$ ， $3$\hspace{1}{x}_{n}$ によるデータの平均値を $3$\bar{x}$ ，分散を $3${\hspace{1}{s}_{x}}^{2}$ ，標準偏差を $3$\hspace{1}{s}_{x}$ とする．各データを $3$a$ 倍し， $3$b$ を加えて得られる $3$n$ 個のデータ $3$\hspace{1}{y}_{1}=a\hspace{1}{x}_{1}+b$ ， $3$\hspace{1}{y}_{2}=a\hspace{1}{x}_{2}+b$ ， $3$\cdots$ ， $3$\hspace{1}{y}_{n}=a\hspace{1}{x}_{n}+b$ を，変量 $3$Y$ のデータとする．変量 $3$Y$ の $3$n$ 個のデータの平均値 $3$\displaystyle{\bar{y}}$ ，分散 $3${ \hspace{1}{s}_{y} }^{2}$ ，標準偏差 $3$\hspace{1}{s}_{y}$ は

$3$\displaystyle{\bar{y}=a\bar{x}+b}$
$3$\displaystyle{\hspace{1}{s}_{y}{ }^{2}={a}^{2}\hspace{1}{s}_{x}{ }^{2}}$
$3$\displaystyle{\hspace{1}{s}_{y}=\left|{a}\right|\hspace{1}{s}_{x}}$

となる．

■導出

●平均値 $3$\displaystyle{\bar{y}}$

$3$\displaystyle{\bar{y}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left({ \hspace{1}{y}_{1}+\hspace{1}{y}_{2}+\cdots +\hspace{1}{y}_{n} }\right)}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left{{ a\left({ \hspace{1}{x}_{1}+\hspace{1}{x}_{2}+\cdots +\hspace{1}{x}_{n} }\right)+nb }\right}}$

$3$\displaystyle{=a\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left({ \hspace{1}{x}_{1}+\hspace{1}{x}_{2}+\cdots +\hspace{1}{x}_{n} }\right)+b}$

$3$\displaystyle{=a\bar{x}+b}$

●分散 $3${ \hspace{1}{s}_{y} }^{2}$

$3$\displaystyle{\hspace{1}{s}_{y}{ }^{2}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left{{ { \left({ \hspace{1}{y}_{1}- \bar{y} }\right) }^{2}+{ \left({ \hspace{1}{y}_{2}- \bar{y} }\right) }^{2}+\cdots +{ \left({ \hspace{1}{y}_{n}- \bar{y} }\right) }^{2} }\right}}$

分散の式を展開し，整理した結果より

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left({ \hspace{1}{y}_{1}{ }^{2}+\hspace{1}{y}_{2}{ }^{2}+\cdots +\hspace{1}{y}_{n}{ }^{2} }\right)- {\left({\bar{y}}\right)}^{2}}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left{{ { \left({ a\hspace{1}{x}_{1}+b }\right) }^{2}+{ \left({ a\hspace{1}{x}_{2}+b }\right) }^{2}+\cdots +{ \left({ a\hspace{1}{x}_{n}+b }\right) }^{2} }\right}}$ $3$\displaystyle{- {\left({ a\bar{x}+b }\right)}^{2}}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left{{ \left({ {a}^{2}\hspace{1}{x}_{1}{ }^{2}+2ab\hspace{1}{x}_{1}+{b}^{2} }\right)+\left({ {a}^{2}\hspace{1}{x}_{2}{ }^{2}+2ab\hspace{1}{x}_{2}+{b}^{2} }\right)+\cdots +\left({ {a}^{2}\hspace{1}{x}_{2}{ }^{2}+2ab\hspace{1}{x}_{n}+{b}^{2} }\right) }\right}}$ $3$\displaystyle{- \left{{ {a}^{2}{ \left({\bar{x}}\right) }^{2}+2ab\bar{x}+{b}^{2} }\right}}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left{{ {a}^{2}\left({ \hspace{1}{x}_{1}{ }^{2}+\hspace{1}{x}_{2}{ }^{2}+\cdots +\hspace{1}{x}_{n}{ }^{2} }\right)+2ab\left({ \hspace{1}{x}_{1}+\hspace{1}{x}_{2}+\cdots +\hspace{1}{x}_{n} }\right)+n{b}^{2} }\right}}$ $3$\displaystyle{- {a}^{2}{\left({\bar{x}}\right)}^{2}- 2ab\bar{x}- {b}^{2}}$

$3$\displaystyle{={a}^{2}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left({ \hspace{1}{x}_{1}{ }^{2}+\hspace{1}{x}_{2}{ }^{2}+\cdots +\hspace{1}{x}_{n}{ }^{2} }\right)}$ $3$\displaystyle{+2ab\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left({ \hspace{1}{x}_{1}+\hspace{1}{x}_{2}+\cdots +\hspace{1}{x}_{n} }\right)}$ $3$\displaystyle{+{b}^{2}- {a}^{2}{\left({\bar{x}}\right)}^{2}- 2ab\bar{x}- {b}^{2}}$

$3$\displaystyle{={a}^{2}\left{{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left({ \hspace{1}{x}_{1}{ }^{2}+\hspace{1}{x}_{2}{ }^{2}+\cdots +\hspace{1}{x}_{n}{ }^{2} }\right)- { \left({\bar{x}}\right) }^{2} }\right}}$

$3$\displaystyle{={a}^{2}\hspace{1}{s}_{x}{ }^{2}}$

●標準偏差 $3$\hspace{1}{s}_{y}$

$3$\displaystyle{\hspace{1}{s}_{y}=\sqrt{ \hspace{1}{s}_{y}{ }^{2} }=\sqrt{ {a}^{2}\hspace{1}{s}_{x}{ }^{2} }=\left|{a}\right|\hspace{1}{s}_{x}}$

によって求められる．

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最終更新日：2026年3月19日