Given a random variableMathworldPlanetmath X, the kth moment of X is the value E[Xk], if the expectation exists.

Note that the expected value is the first moment of a random variable, and the varianceMathworldPlanetmath is the second moment minus the first moment squared.

The kth moment of X is usually obtained by using the moment generating function.

Given a random variable X, the kth central moment of X is the value E[(X-E[X])k], if the expectation exists. It is denoted by μk.

Note that the μ1=0 and μ2=Var[X]=σ2. The third central moment divided by the standard deviationMathworldPlanetmath cubed is called the skewness τ:


The skewness measures how “symmetrical”, or rather, how “skewed”, a distributionPlanetmathPlanetmathPlanetmath is with respect to its mode. A non-zero τ means there is some degree of skewness in the distribution. For example, τ>0 means that the distribution has a longer positivePlanetmathPlanetmath tail.

The fourth central moment divided by the fourth power of the standard deviation is called the kurtosis κ:


The kurtosis measures how “peaked” a distribution is compared to the standard normal distributionMathworldPlanetmath. The standard normal distribution has κ=3. κ<3 means that the distribution is “flatter” than then standard normal distribution, or platykurtic. On the other hand, a distribution with κ>3 can be characterized as being more “peaked” than N(0,1), or leptokurtic.

Title moment
Canonical name Moment
Date of creation 2013-03-22 11:53:54
Last modified on 2013-03-22 11:53:54
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 60-00
Classification msc 62-00
Classification msc 81-00
Defines central moment
Defines skewness
Defines kurtosis
Defines platykurtic
Defines leptokurtic