Standardized moment
# 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 X be a random variable with a probability distribution P and mean value μ=E[X]{\textstyle \mu =\mathrm {E} [X]} (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 ${\frac {\mu _{k}}{\sigma ^{k}}},$ that is, the ratio of the kth moment about the mean

    $\mu _{k}=\operatorname {E} \left[(X-\mu )^{k}\right]=\int _{-\infty }^{\infty }(x-\mu )^{k}P(x)\,dx,$

to the kth power of the standard deviation,

    $\sigma ^{k}=\mu _{2}^{k/2}=\left({\sqrt {\mathrm {E} \left[(X-\mu )^{2}\right]}}\right)^{k}.$

The power of k is because moments scale as $x^{k},$ meaning that $\mu _{k}(\lambda X)=\lambda ^{k}\mu _{k}(X):$ 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 | ${\tilde {\mu }}_{1}={\frac {\mu _{1}}{\sigma ^{1}}}={\frac {\operatorname {E} \left[(X-\mu )^{1}\right]}{(\operatorname {E} \left[(X-\mu )^{2}\right])^{1/2}}}={\frac {\mu -\mu }{\sqrt {\operatorname {E} \left[(X-\mu )^{2}\right]}}}=0$ | The first standardized moment is zero, because the first moment about the mean is always zero. |
| 2 | ${\tilde {\mu }}_{2}={\frac {\mu _{2}}{\sigma ^{2}}}={\frac {\operatorname {E} \left[(X-\mu )^{2}\right]}{(\operatorname {E} \left[(X-\mu )^{2}\right])^{2/2}}}=1$ | The second standardized moment is one, because the second moment about the mean is equal to the variance $σ^2.$ |
| 3 | ${\tilde {\mu }}_{3}={\frac {\mu _{3}}{\sigma ^{3}}}={\frac {\operatorname {E} \left[(X-\mu )^{3}\right]}{(\operatorname {E} \left[(X-\mu )^{2}\right])^{3/2}}}$ | The third standardized moment is a measure of skewness. |
| 4 | ${\tilde {\mu }}_{4}={\frac {\mu _{4}}{\sigma ^{4}}}={\frac {\operatorname {E} \left[(X-\mu )^{4}\right]}{(\operatorname {E} \left[(X-\mu )^{2}\right])^{4/2}}}$ | 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

Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation, ${\frac {\sigma }{\mu }}$. 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.

