Standardized moment

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In probability theory and statistics, the kth standardized moment of a probability distribution is $\frac{\mu_k}{\sigma^k}\!$ where $μ k$ 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 $μ k (λ X) = λ k μ k (X)$ : 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, 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 (the square of the standard deviation)
The third standardized moment is the skewness
The fourth standardized moment is the kurtosis

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

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.