Standardized Moment

In probability theory and statistics, the kth standardized moment of a probability distribution is where 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, meaning that : 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 kurtosis

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