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Standardized moment

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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

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Let X be a random variable with a probability distribution P and mean value (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 ,[2] that is, the ratio of the k-th moment about the mean

to the k-th power of the standard deviation,

The power of k is because moments scale as , meaning that 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 The first standardized moment is zero, because the first moment about the mean is always zero.
2 The second standardized moment is one, because the second moment about the mean is equal to the variance σ2.
3 The third standardized moment is a measure of skewness.
4 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.

Other normalizations

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Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation, . However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because 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.

See also

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References

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  1. ^ Ramsey, James Bernard; Newton, H. Joseph; Harvill, Jane L. (2002-01-01). "CHAPTER 4 MOMENTS AND THE SHAPE OF HISTOGRAMS". The Elements of Statistics: With Applications to Economics and the Social Sciences. Duxbury/Thomson Learning. p. 96. ISBN 9780534371111.
  2. ^ Weisstein, Eric W. "Standardized Moment". mathworld.wolfram.com. Retrieved 2016-03-30.