## 5.1 Central Moments

Central Moments: $$\mu_r=\frac{\sum(x_i-\bar x)^r}{n}$$

• $$\mu_1=\frac{\sum(x_i-\bar x)}{n}=\frac{\sum x_i}{n}-\frac{n \bar x}{n}=\frac{n \bar x}{n}-\frac{n \bar x}{n}=0$$

• $$\mu_2=\frac{\sum(x_i-\bar x)^2}{n}=\sigma^2$$

• $$\mu_3=\frac{\sum(x_i-\bar x)^3}{n}$$

• $$\mu_4=\frac{\sum(x_i-\bar x)^4}{n}$$

• For grouped data: $$\mu_r=\frac{\sum f_i(x_i-\bar x)^3}{n}$$