統計学の基本的な定義

n個のデータ x1, x2, x3, ... xnが得られたとき、これらの標本の要約を行うための種々の定義列挙します。

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Mean : $\bar{x}$
\begin{eqnarray}
\bar{x} = \frac{x_1 + x_2 + \cdots + x_n} {n} =\sum^n_{i=1}x_i
\end{eqnarray}

Median : $\widetilde{x}$
\begin{eqnarray}
\widetilde{x}=\left\{ \begin{array}{ll}
the \, middle \, number \\
the\, mean\,  of\, the\, pair\, of\, middle\, numbers \\
\end{array} \right.
\end{eqnarray}

Sum of squares : $S$
\begin{eqnarray}
S = \sum^n_{i+1}(x_i - \widetilde{x})^2 = \sum^n_{i=1}x_i^2 -\frac{(\sum^n_{i=1}x_i)^2}{n}
\end{eqnarray}

Unbiased variance : $V$
\begin{eqnarray}
V = = S/(n-1)=\sum^n_{i+1}((x_i - \bar{x})^2)^2 / (n - 1)
\end{eqnarray}

Standard deviation : $S.D.$
\begin{eqnarray}
S.D = \sqrt{V}
\end{eqnarray}

Range : $R$
\begin{eqnarray}
R = x_{max} - x_{min}
\end{eqnarray}

quantile deviation : $Q.D$
\begin{eqnarray}
Q.D = \frac{Q_3-Q_1}{2}
\end{eqnarray}

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