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 <math display="inline">\mu = \operatorname{E}[X]</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
<math display="block"> \mu_k = \operatorname{E} \left[ ( X - \mu )^k \right] = \int_{-\infty}^{\infty} {\left(x - \mu\right)}^k f(x)\,dx, </math>
to the -th power of the standard deviation,
<math display="block">\sigma^k = \mu_2^{k/2} = \operatorname{E}\!{\left[ {\left(X - \mu\right)}^2 \right]}^{k/2}.</math>
The power of is because moments scale as meaning that <math>\mu_k(\lambda X) = \lambda^k \mu_k(X):</math> 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:
{| class="wikitable"
!Degree k
!
!Comment
|-
|1
|<math>
\alpha_1 = \frac{\mu_1}{\sigma^1}= \frac{\operatorname{E} \left[ ( X - \mu )^1 \right]}{\left( \operatorname{E} \left[ ( X - \mu )^2 \right]\right)^{1/2
= \frac{\mu - \mu}{\sqrt{ \operatorname{E} \left[ ( X - \mu )^2 \right]
= 0
</math>
|The first standardized moment is zero, because the first moment about the mean is always zero.
|-
|2
|<math>
\alpha_2 = \frac{\mu_2}{\sigma^2}
= \frac{\operatorname{E} \left[ ( X - \mu )^2 \right]}{\left( \operatorname{E} \left[ ( X - \mu )^2 \right]\right)^{2/2
= 1
</math>
|The second standardized moment is one, because the second moment about the mean is equal to the variance .
|-
|3
|<math>
\alpha_3 = \frac{\mu_3}{\sigma^3}
= \frac{\operatorname{E} \left[ ( X - \mu )^3 \right]}{\left( \operatorname{E} \left[ ( X - \mu )^2 \right]\right)^{3/2
</math>
|The third standardized moment is a measure of skewness.
|-
|4
|<math>
\alpha_4 = \frac{\mu_4}{\sigma^4}
= \frac{\operatorname{E} \left[ ( X - \mu )^4 \right]}{\left( \operatorname{E} \left[ (X - \mu)^2 \right]\right)^{4/2
</math>
|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, <math>\sigma / \mu</math>. However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because <math>\mu</math> 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