Standardized Moment

Defined for samples $x_i$, $i=1$, ..., $N$ by

$$\alpha_r \equiv {1\over N} \sum_{i=1}^N {z_i}^r = {\mu_r\over\sigma^r},$$ (1)

where

$$z_i\equiv {x_i-\bar x\over s_x}.$$ (2)

The first few are

$$\alpha_1 = 0$$ (3)

$$\alpha_2 = 1$$ (4)

$$\alpha_3 = {\mu_3\over s^3}$$ (5)

$$\alpha_4 = {\mu_4\over s^4}.$$ (6)

See also Kurtosis, Moment, Skewness