【定理・公式・証明】高校数学定理・公式 – 数学Ⅰ – 平均・分散・共分散・標準偏差における変量の変換

$x$ の平均値 $\overline{x}$ は， 

$\overline{x} = \frac{1}{n} (x_{1} + x_{2} + \cdots + x_{n})$

このとき， $u_{i}$ の平均値 $\overline{u}$ は，

$\overline{u} = \frac{1}{n} (u_{1} + u_{2} + \cdots + u_{n})$

$= \frac{1}{n} \{ (ax_{1} +b) + (ax_{2} +b) + \cdots + (ax_{n} +b) \}$

$= \frac{1}{n} \{ a (x_{1} + x_{2} + \cdots + x_{n} ) +nb \}$

$= a \cdot \frac{1}{n} (x_{1} + x_{2} + \cdots + x_{n} ) +b$

$=a \overline{x} +b$

$x$ の平均値 $\overline{x}$ ，分散 $s_{x}^2$ は，

$\begin{eqnarray} \left\{ \begin{array}{l} \displaystyle \overline{x} = \frac{1}{n} (x_{1} + x_{2} + x_{n} ) \\ \displaystyle s_{x}^2= \frac{1}{n} \{ (x_{1} – \overline{x} )^2 + (x_{2} – \overline{x} )^2 + \cdots + (x_{n} – \overline{x} ) ^2 \} \ ( \gt 0 ) \end{array} \right. \end{eqnarray}$

$u_{i} =ax_{i} +b$ のとき， $\overline{u} =a \overline{x} +b$ 。このとき，$u$ の分散 $u_{x}^2$ は，

$u_{1} – \overline{u} =ax_{1} +b-(a \overline{x} +b) = a(x_{1} – \overline{x}$

$u_{2} – \overline{u} =ax_{2} +b-(a \overline{x} +b) = a(x_{2} – \overline{x}$

$\vdots$

$u_{n} – \overline{u} =ax_{n} +b-(a \overline{x} +b) = a(x_{n} – \overline{x}$

より，

[Ⅰ] $\displaystyle s_{u}^2 = \frac{1}{n} \{ (u_{1} – \overline{u} )^2 + (u_{2} – \overline{u} )^2 + \cdots + ( u_{n} – \overline{u} )^2 \}$

$\displaystyle = \frac{1}{n} \{ a^2 (x_{1} – \overline{x} )^2 + a^2 (x_{2} – \overline{x} )^2 + \cdots + a^2 (x_{n} – \overline{x} )^2 \} $

$\displaystyle =a^2 \cdot \frac{1}{n} \{ ( x_{1} – \overline{x} )^2 + (x_{2} – \overline{x} )^2 + \cdots +(x_{n} – \overline{x} )^2 \} = a^2 \cdot s_{x} ^2$

したがって，

[Ⅱ] $s_{u} = \sqrt{a^2 \cdot s_{x}^2} = \vert a \vert s_{x}$

$x_{i}$ , $y_{i}$ の共分散 $s_{xy}$ は，

$\displaystyle s_{xy} = {1}{n} \{ (x_{1} – \overline{x} )(y_{1} – \overline{y} ) + \cdots + (x_{n} – \overline{x} )( y_{n} – \overline{y} ) \}$

このとき， $u_{i}$ ， $v_{i}$ の共分散 $s_{uv}$ は，

$\displaystyle s_{uv} = \frac{1}{n} \{ (u_{1} – \overline{u} )(v_{1} – \overline{v} )+( u_{2} – \overline{u} )( v_{2} – \overline{v} ) + \cdots + u_{n} – \overline{u} )( v_{n} – \overline{v} ) \}$

$\displaystyle = \frac{1}{n} ( \{ ax_{1} +b- (a \overline{x} +b) \} \{ cy_{1} +d-(c \overline{y} +d) \} + \cdots + \{ ax_{n} +b-(a \overline{x} + b) \} \{ cy_{n} +d-(c \overline{y} +d) \} )$

$\displaystyle = \frac{1}{n} \{ a (x_{1} -\overline{x} ) \cdot c( y_{1} – \overline{y} ) + \cdots +a (x_{n} – \overline{x} ) \cdot c ( y_{n} – \overline{y} ) \}$

$\displaystyle =a \cdot c \cdot \frac{1}{n} \{ x_{1} – \overline{x} )(y_{1} – \overline{y} )+ \cdots + ( x_{n} – \overline{x} )( y_{n} – \overline{y} ) \}$

$=acs_{xy}$