Standardized moment
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 (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
[edit]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
[edit]References
[edit]- ^ 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.
- ^ Weisstein, Eric W. "Standardized Moment". mathworld.wolfram.com. Retrieved 2016-03-30.