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

Sources

• Weisstein, Eric W. "Standardized Moment." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StandardizedMoment.html