# 偏度

## 介紹

• 負偏態左偏態：左側的尾部更長，分布的主體集中在右側。[2]
• 正偏態右偏態：右側的尾部更長，分布的主體集中在左側。[2]

## 定義

${\displaystyle \gamma _{1}=\operatorname {E} {\Big [}{\big (}{\tfrac {X-\mu }{\sigma }}{\big )}^{\!3}\,{\Big ]}={\frac {\mu _{3}}{\sigma ^{3}}}={\frac {\operatorname {E} {\big [}(X-\mu )^{3}{\big ]}}{\ \ \ (\operatorname {E} {\big [}(X-\mu )^{2}{\big ]})^{3/2}}}={\frac {\kappa _{3}}{\kappa _{2}^{3/2}}}\ ,}$

${\displaystyle \gamma _{1}=\operatorname {E} {\bigg [}{\Big (}{\frac {X-\mu }{\sigma }}{\Big )}^{\!3}\,{\bigg ]}={\frac {\operatorname {E} [X^{3}]-3\mu \operatorname {E} [X^{2}]+2\mu ^{3}}{\sigma ^{3}}}={\frac {\operatorname {E} [X^{3}]-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}\ .}$

## 樣本偏度

${\displaystyle g_{1}={\frac {m_{3}}{{m_{2}}^{3/2}}}={\frac {{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{3}}{\left({\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}\right)^{3/2}}}\ ,}$

### 性質

${\displaystyle \Pr[X>x]={\frac {(1+x)^{-3}}{2}}}$ ${\displaystyle x}$ 為正）時，偏度無法定義。

${\displaystyle \Pr \left[X>x\right]=x^{-2}{\mbox{ for }}x>1,\ \Pr[X<1]=0}$

## 註釋

1. 存档副本. [2018-12-14]. （原始内容存档于2020-11-12）.
2. 存档副本. [2010-10-30]. （原始内容存档于2011-08-11）.

## 參考資料

• Groeneveld, RA; Meeden, G. Measuring Skewness and Kurtosis. The Statistician. 1984, 33 (4): 391–399 [2010-10-30]. doi:10.2307/2987742. （原始内容存档于2020-08-20）.
• Johnson, NL, Kotz, S, Balakrishnan N (1994) Continuous Univariate Distributions, Vol 1, 2nd Edition Wiley ISBN 0-471-58495-9
• MacGillivray, HL. Shape properties of the g- and h- and Johnson families. Comm. Statistics - Theory and Methods. 1992, 21: 1244–1250.