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]
Let X be a random variable with a probability distribution P and mean value [math]\displaystyle{ \mu = \mathrm{E}[X] }[/math] (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 [math]\displaystyle{ \frac{\mu_k}{\sigma^k}, }[/math][2] that is, the ratio of the kth moment about the mean
to the kth power of the standard deviation,
The power of k is because moments scale as [math]\displaystyle{ x^k, }[/math] meaning that [math]\displaystyle{ \mu_k(\lambda X) = \lambda^k \mu_k(X): }[/math] 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:
Degree k | Comment | |
---|---|---|
1 | [math]\displaystyle{ \tilde{\mu}_1 = \frac{\mu_1}{\sigma^1} = \frac{\operatorname{E} \left[ ( X - \mu )^1 \right]}{( \operatorname{E} \left[ ( X - \mu )^2 \right])^{1/2}} = \frac{\mu - \mu}{\sqrt{ \operatorname{E} \left[ ( X - \mu )^2 \right]}} = 0 }[/math] | The first standardized moment is zero, because the first moment about the mean is always zero. |
2 | [math]\displaystyle{ \tilde{\mu}_2 = \frac{\mu_2}{\sigma^2} = \frac{\operatorname{E} \left[ ( X - \mu )^2 \right]}{( \operatorname{E} \left[ ( X - \mu )^2 \right])^{2/2}} = 1 }[/math] | The second standardized moment is one, because the second moment about the mean is equal to the variance σ2. |
3 | [math]\displaystyle{ \tilde{\mu}_3 = \frac{\mu_3}{\sigma^3} = \frac{\operatorname{E} \left[ ( X - \mu )^3 \right]}{( \operatorname{E} \left[ ( X - \mu )^2 \right])^{3/2}} }[/math] | The third standardized moment is a measure of skewness. |
4 | [math]\displaystyle{ \tilde{\mu}_4 = \frac{\mu_4}{\sigma^4} = \frac{\operatorname{E} \left[ ( X - \mu )^4 \right]}{( \operatorname{E} \left[ ( X - \mu )^2 \right])^{4/2}} }[/math] | The fourth standardized moment refers to the kurtosis. |
For skewness and kurtosis, alternative definitions exist, which are based on the third and fourth cumulant respectively.
Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation, [math]\displaystyle{ \frac{\sigma}{\mu} }[/math]. However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because [math]\displaystyle{ \mu }[/math] 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.
Original source: https://en.wikipedia.org/wiki/Standardized moment.
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