Standardized moment

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{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} {{ safesubst:#invoke:Unsubst||$N=Technical |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In probability theory and statistics, the kth standardized moment of a probability distribution is where is the kth moment about the mean and σ is the standard deviation.

It is the normalization of the kth moment with respect to standard deviation. The power of k is because moments scale as , meaning that : they are homogeneous polynomials 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 standardized moment is zero, because the first moment about the mean is zero
  • The second standardized moment is one, because the second moment about the mean is equal to the variance (the square of the standard deviation)
  • The third standardized moment is the skewness
  • The fourth standardized moment is the kurtosis

Note that for skewness and kurtosis alternative definitions exist, which are based on the third and fourth cumulant respectively.

Other normalizations

Template:Rellink 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