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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and statistics, a standardized moment of a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
is a moment (often a higher degree
central moment In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
) that is normalized, typically by a power of the standard deviation, rendering the moment
scale invariant In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term ...
. The
shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on ...
of different probability distributions can be compared using standardized moments.


Standard normalization

Let ''X'' be a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
with a probability distribution ''P'' and mean value \mu = \mathrm /math> (i.e. the first raw moment or moment about zero), the operator E denoting the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of ''X''. Then the standardized moment of degree ''k'' is \frac, that is, the ratio of the ''k''th moment about the mean : \mu_k = \operatorname \left ( X - \mu )^k \right = \int_^ (x - \mu)^k P(x)\,dx, to the ''k''th power of the standard deviation, :\sigma^k = \left(\sqrt\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 function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the '' ...
s of degree ''k'', thus the standardized moment is
scale invariant In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term ...
. 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 number A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
s. 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 In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will hav ...
respectively.


Other normalizations

Another scale invariant, dimensionless measure for characteristics of a distribution is the
coefficient of variation In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed ...
, \frac. 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) In statistics and applications of statistics, normalization can have a range of meanings. In the simplest cases, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averagin ...
for further normalizing ratios.


See also

*
Coefficient of variation In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed ...
* Moment (mathematics) *
Central moment In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
*


References

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