平均、分散、標準偏差

平均：\( \bar{x} = \displaystyle \frac{1}{n} (x_1 + x_2 + x_3 + \ldots + x_x) \)
分散：\( {S_x}^2 = \displaystyle \frac{1}{n} \{ (x_1 – \bar{x})^2 + (x_2 – \bar{x})^2 + (x_3 – \bar{x})^2 + \cdots +(x_n – \bar{x})^2 \} = \bar{x^2} – (\bar{x})^2 \)
標準偏差：\( S_x = \sqrt{ {S_x}^2 } \)

---

分散：\( {S_x}^2 = \bar{x^2} – (\bar{x})^2 \)

\( \begin{eqnarray}
{S_x}^2 &=& \displaystyle \frac{1}{n} \{ (x_1 – \bar{x})^2 + (x_2 – \bar{x})^2 + (x_3 – \bar{x})^2 + \cdots +(x_n – \bar{x})^2 \} \\
\
&=& \displaystyle \frac{1}{n} \{ ({x_1}^2 – 2 x_1 \bar{x} + \bar{x}^2) + ({x_2}^2 – 2 x_2 \bar{x} + \bar{x}^2) + ({x_3}^2 – 2 x_3 \bar{x} + \bar{x}^2) + \cdots + ({x_n}^2 – 2 x_b \bar{x} + \bar{x}^2) \} \\
\
&=& \displaystyle \frac{1}{n} ({x_1}^2 + {x_2}^2 + {x_3}^2 + \cdots + {x_n}^2) \
– \displaystyle 2 \cdot \frac{1}{n} (x_1 + x_2 + x_3 + \cdots + x_n) \cdot \bar{x} \
+ \displaystyle \frac{n}{n} \cdot\bar{x}^2 \\
\
&=& \bar{x^2} – (\bar{x})^2
\end{eqnarray} \)

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\( y_i = ax_i + b \qquad (a, b は定数、i = 1, 2, 3, \cdots ,n) \) のとき

平均 \( \bar{y} = a\bar{x} + b \)

\( \begin{eqnarray}
\bar{y} &=& \displaystyle \frac{1}{n} \{ (ax_1 + b) + (ax_2 + b) + (ax_3 + b) + \cdots + (an_1 + b) \} \\
&=& \displaystyle \frac{1}{n} \{ a(x_1 + x_2 + x_3 + \cdots + x_n) + nb \} \\
&=& a \bar{x} + b
\end{eqnarray} \)

分散 \( {S_y}^2 = a^2 \cdot {S_x}^2 \)

\( \begin{eqnarray}
{S_y}^2 &=& \displaystyle \frac{1}{n} [ \{ (ax_1 + b) – (a \bar{x} + b) \}^2 + \{ (ax_2 + b) – (a \bar{x} + b) \}^2 + \{ (ax_3 + b) – (a \bar{x} + b) \}^2 + \cdots + \{ (ax_n + b) – (a \bar{x} + b) \}^2 ] \\
&=& \displaystyle \frac{1}{n} \{ a^2 (x_1 – \bar{x})^2 + a^2 (x_2 – \bar{x})^2 + a^2 (x_3 – \bar{x})^2 + \cdots + a^2 (x_n – \bar{x})^2 \} \\
&=& a^2 \cdot \displaystyle \frac{1}{n} \{ (x_1 – \bar{x})^2 + (x_2 – \bar{x})^2 + (x_3 – \bar{x})^2 + \cdots +(x_n – \bar{x})^2 \} \\
&=& a^2 \cdot {S_x}^2
\end{eqnarray} \)

標準偏差 \( S_y = \vert a \vert S_x \)

\( \begin{eqnarray}
S_y &=& \sqrt{ a^2 \cdot {S_x}^2 } \\
&=& \vert a \vert S_x
\end{eqnarray} \)