In
probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
, the
kth
standardized moment of a
probability distributionIn probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval...
is where is the
kth moment about the mean and σ is the
standard deviationIn probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance. Standard deviation is a widely used measure of the variability or dispersion, being algebraically more tractable though...
.
It is the
normalizationIn one usage in statistics, normalization is the process of removing statistical error in repeated measured data. A normalization is sometimes based on a property...
of the
kth moment with respect to
standard deviationIn probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance. Standard deviation is a widely used measure of the variability or dispersion, being algebraically more tractable though...
. The power of
k is because moments scale as , meaning that : they are
homogeneous polynomialIn mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, is a homogeneous polynomial...
s of degree
k, thus the standardized moment is scale invariant. This can also be understood as being because moments have dimension, but in the ratio defining standardized moments, the dimensions cancel, so they are dimensionless numbers.
- The first standardized moment is zero, because the first moment about the mean is zero
- The second standardized moment is one, because the second moment about the mean is equal to the variance
In probability theory and statistics, the variance of a random variable or distribution is the expected square deviation of that variable from its expected value or mean, or to put it another way: variance is the measure of the amount of variation of all the scores for a variable...
(the square of the standard deviation)
- The third standardized moment is the skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.-Introduction :...
- The fourth standardized moment is the kurtosis
In probability theory and statistics, kurtosis is a measure of the "peakedness" of the probability distribution of a real-valued random variable...
Note that for skewness and kurtosis alternative definitions exist, which are based on the third and fourth
cumulantIn probability theory and statistics, the cumulants κn of a random variable X are defined by the cumulant-generating function, the logarithm of the moment-generating function, if it exists:...
respectively.
Another scale invariant, dimensionless measure for characteristics of a distribution is the
coefficient of variationIn probability theory and statistics, the coefficient of variation is a normalized measure of dispersion of a probability distribution. It is defined as the ratio of the standard deviation to the mean :...
, .
In
probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
, the
kth
standardized moment of a
probability distributionIn probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval...
is where is the
kth moment about the mean and σ is the
standard deviationIn probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance. Standard deviation is a widely used measure of the variability or dispersion, being algebraically more tractable though...
.
It is the
normalizationIn one usage in statistics, normalization is the process of removing statistical error in repeated measured data. A normalization is sometimes based on a property...
of the
kth moment with respect to
standard deviationIn probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance. Standard deviation is a widely used measure of the variability or dispersion, being algebraically more tractable though...
. The power of
k is because moments scale as , meaning that : they are
homogeneous polynomialIn mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, is a homogeneous polynomial...
s of degree
k, thus the standardized moment is scale invariant. This can also be understood as being because moments have dimension, but in the ratio defining standardized moments, the dimensions cancel, so they are dimensionless numbers.
- The first standardized moment is zero, because the first moment about the mean is zero
- The second standardized moment is one, because the second moment about the mean is equal to the variance
In probability theory and statistics, the variance of a random variable or distribution is the expected square deviation of that variable from its expected value or mean, or to put it another way: variance is the measure of the amount of variation of all the scores for a variable...
(the square of the standard deviation)
- The third standardized moment is the skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.-Introduction :...
- The fourth standardized moment is the kurtosis
In probability theory and statistics, kurtosis is a measure of the "peakedness" of the probability distribution of a real-valued random variable...
Note that for skewness and kurtosis alternative definitions exist, which are based on the third and fourth
cumulantIn probability theory and statistics, the cumulants κn of a random variable X are defined by the cumulant-generating function, the logarithm of the moment-generating function, if it exists:...
respectively.
Other normalizations
Another scale invariant, dimensionless measure for characteristics of a distribution is the
coefficient of variationIn probability theory and statistics, the coefficient of variation is a normalized measure of dispersion of a probability distribution. It is defined as the ratio of the standard deviation to the mean :...
, . However, this is not a standardized moment, firstly because 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)In one usage in statistics, normalization is the process of removing statistical error in repeated measured data. A normalization is sometimes based on a property...
for further normalizing ratios.
See also
- Coefficient of variation
In probability theory and statistics, the coefficient of variation is a normalized measure of dispersion of a probability distribution. It is defined as the ratio of the standard deviation to the mean :...
- Moment (mathematics)
The concept of moment in mathematics evolved from the concept of moment in physics. The nth moment of a real-valued function f of a real variable about a value c is...
- Standard score: Other normalizations