Definition:Standardized Moment

Definition
Let $X$ be a random variable on some probability space with standard deviation $\sigma$.

Then the $n$th standardized moment of $X$, usually denoted $\alpha_n$, is defined as:


 * $\alpha_n = \dfrac {\mu_n} {\sigma^n}$

where $\mu_n$ is the $n$th central moment of $X$.

The third standardized moment of $X$ is the skewness of $X$, and is usually denoted $\gamma_1$.

The fourth standardized moment of $X$ is the kurtosis of $X$, and is usually denoted $\alpha_4$.

Also see

 * Definition:Moment (Probability Theory)
 * Definition:Raw Moment
 * Definition:Central Moment