# 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。[1]
• 正偏態右偏態：右側的尾部更長，分布的主體集中在左側。[1]

## 定義

$\gamma_1 = \operatorname{E}\Big[\big(\tfrac{X-\mu}{\sigma}\big)^{\!3}\, \Big] = \frac{\mu_3}{\sigma^3} = \frac{\operatorname{E}\big[(X-\mu)^3\big]}{\ \ \ ( \operatorname{E}\big[ (X-\mu)^2 \big] )^{3/2}} = \frac{\kappa_3}{\kappa_2^{3/2}}\ ,$

$\gamma_1 = \operatorname{E}\bigg[\Big(\frac{X-\mu}{\sigma}\Big)^{\!3} \,\bigg] = \frac{\operatorname{E}[X^3] - 3\mu\operatorname E[X^2] + 2 \mu^3}{\sigma^3} = \frac{\operatorname{E}[X^3] - 3\mu\sigma^2 - \mu^3}{\sigma^3}\ .$

## 樣本偏度

$g_1 = \frac{m_3}{{m_2}^{3/2}} = \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^3}{\left(\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^2\right)^{3/2}}\ ,$

### 性質

$\Pr[X>x]=(1+x)^{-3}/2$（x為正）時，偏度無法定義。

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

## 參考資料

• Groeneveld, RA; Meeden, G. Measuring Skewness and Kurtosis. The Statistician. 1984, 33 (4): 391–399. doi:10.2307/2987742.
• Johnson, NL, Kotz, S, Balakrishnan N (1994) Continuous Univariate Distributions, Vol 1, 2nd Edition Wiley ISBN0-471-58495-9
• MacGillivray, HL. Shape properties of the g- and h- and Johnson families. Comm. Statistics - Theory and Methods. 1992, 21: 1244–1250.