Standardized moment

In probability theory and statistics, the kth standardized moment of a probability distribution is $\frac{\mu_k}{\sigma^k}\!$ where $\mu_k$ is the kth moment about the mean and σ is the standard deviation.

It is the normalization of the kth moment with respect to standard deviation. 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 polynomials 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 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 (the square of the standard deviation)
The third standardized moment is the skewness
The fourth standardized moment is the historical kurtosis (used in older works)

Note that for skewness and kurtosis alternative definitions exist, which are based on the third and fourth cumulant respectively.

Other normalizations

For more details on this topic, see Normalization (statistics).

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.

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Standardized moment

Other normalizations

See also