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.

Standard normalization

Let be a random variable with a probability distribution and mean value \mu = \operatorname /math> (i.e. the first raw moment or moment about zero), the operator denoting the expected value of . Then the standardized moment of degree is that is, the ratio of the -th moment about the mean

\mu_k = \operatorname \left ( X - \mu )^k \right = \int_^ ^k f(x)\,dx,

to the -th power of the standard deviation,

$\sigma^k = \mu_2^ = \operatorname\!^.$

The power of is because moments scale as meaning that $\mu_k(\lambda X) = \lambda^k \mu_k(X):$ they are homogeneous functions of degree , 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

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

See also

*Coefficient of variation
*Moment (mathematics)
*Central moment
*

References

{{DEFAULTSORT:Standardized Moment
Statistical deviation and dispersion
Statistical ratios
Moments (mathematics)