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: \(\mu_r=\frac{\sum f_i(x_i-\bar x)^3}{n}\)