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Normalized central moments
In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. The shape of different probability distributions can be compared using standardized moments.[1]
Standard normalization[edit]Let X be a random variable with a probability distribution P and mean value μ = E [ X ] {\textstyle \mu =\operatorname {E} [X]} (i.e. the first raw moment or moment about zero), the operator E denoting the expected value of X. Then the standardized moment of degree k is μ k / σ k {\displaystyle \mu _{k}/\sigma ^{k}} ,[2] that is, the ratio of the k-th moment about the mean
μ k = E [ ( X − μ ) k ] = ∫ − ∞ ∞ ( x − μ ) k f ( x ) d x , {\displaystyle \mu _{k}=\operatorname {E} \left[(X-\mu )^{k}\right]=\int _{-\infty }^{\infty }{\left(x-\mu \right)}^{k}f(x)\,dx,}
to the k-th power of the standard deviation,
σ k = μ 2 k / 2 = E [ ( X − μ ) 2 ] k / 2 . {\displaystyle \sigma ^{k}=\mu _{2}^{k/2}=\operatorname {E} \!{\left[{\left(X-\mu \right)}^{2}\right]}^{k/2}.}
The power of k is because moments scale as x k {\displaystyle x^{k}} , meaning that μ k ( λ X ) = λ k μ k ( X ) : {\displaystyle \mu _{k}(\lambda X)=\lambda ^{k}\mu _{k}(X):} they are homogeneous functions of degree k, thus the standardized moment is scale invariant. This can also be understood as being because moments have dimension; in the above ratio defining standardized moments, the dimensions cancel, so they are dimensionless numbers.
The first four standardized moments can be written as:
For skewness and kurtosis, alternative definitions exist, which are based on the third and fourth cumulant respectively.
Other normalizations[edit]Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation, σ / μ {\displaystyle \sigma /\mu } . However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because μ {\displaystyle \mu } is the first moment about zero (the mean), not the first moment about the mean (which is zero).
See Normalization (statistics) for further normalizing ratios.
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