平均差、方差与标准差

平均差

$$A.D. = \frac{\sum{\vert X_{i} - \bar{X} \vert}}{n}$$

方差

$$s^{2} = \frac{\sum{(X_{i} - \bar{X})^{2}}}{n} = \frac{\sum{X_{i}^{2}}}{n} - \bigg(\frac{\sum{X_{i}}}{n}\bigg)^{2}$$

方差的可加性

$$\begin{array}{ll} \because & S^{2}_{T} = \frac{1}{n} \sum_{i=1}^{n}{(X_{i} - \bar{X}_{T})^{2}} \\ \therefore & S_{T}^{2} = \frac{1}{n} \sum_{j=1}^{k}\bigg[\sum_{i=1}^{n_{j}}(X_{i} - \bar{X}_{T})^{2}\bigg] \\ & = \frac{1}{n}\sum_{j=1}^{k}\bigg[\sum_{i=1}^{n_{j}}(X_{i} - \bar{X}_{j} + \bar{X}_{j} - \bar{X}_{T})^{2}\bigg] \\ & = \frac{1}{n}\sum_{j=1}^{k}\bigg[\sum_{i=1}^{n_{j}}(X_{i} - \bar{X}_{j})^{2} + \sum_{i=1}^{n_{j}}(\bar{X}_{j} - \bar{X}_{T})^{2}\bigg] \\ & = \frac{1}{n}\sum_{j=1}^{k}\bigg[n_{j}\cdot s_{j}^{2} + n_{j} \cdot (\bar{X}_{j} - \bar{X}_{T})^{2} \bigg] \\ \therefore & \color{#D0104C}{S_{T}^{2} = \frac{\sum_{j=1}^{k}{n_{j}\cdot s_{j}^{2}} + \sum_{j=1}^{k}n_{j}\cdot(\bar{X}_{j} - \bar{X}_{T})^{2}}{\sum_{j=1}^{k}n_{j}}} \\ \end{array}$$

标准差

$$s = \sqrt{s^{2}}$$

特性

• $Y_{i} = X_{i} + C \Rightarrow s_{Y} = s_{X}$
• $Y_{i} = C\cdot X_{i} \Rightarrow s_{Y} = \vert C \vert\cdot s_{X}$
• $Y_{i} = C\cdot X_{i} + d\Rightarrow s_{Y} = \vert C \vert\cdot s_{X}$

优点

• 反应灵敏
• 计算严密
• 计算简单
• 简单明了，易于理解
• 适合于进一步用代数方法演算
• 较少受抽样变动的影响

缺点

• 运算繁琐
• 易受极端数据的影响
• 不易理解

补充

• 切比雪夫定理（“正负三个标准差法则”）