Central Moment - Univariate Moments

Univariate Moments

The kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity μ_k := E)k], where E is the expectation operator. For a continuous univariate probability distribution with probability density function f(x) the moment about the mean μ is

For random variables that have no mean, such as the Cauchy distribution, central moments are not defined.

The first few central moments have intuitive interpretations:

• The "zeroth" central moment μ₀ is one.
• The first central moment μ₁ is zero (not to be confused with the first moment itself, the expected value or mean).
• The second central moment μ₂ is called the variance, and is usually denoted σ2, where σ represents the standard deviation.
• The third and fourth central moments are used to define the standardized moments which are used to define skewness and kurtosis, respectively.