Standardized Moment

A standardized moment is a normalized central moment of a probability distribution, constructed by dividing a central moment by an appropriate power of the standard deviation so that the result is dimensionless and unaffected by the scale of the underlying measurements. Formally, the k-th standardized moment equals the k-th central moment divided by the standard deviation raised to the k-th power. The first standardized moment is always zero, since a distribution is centered at its own mean, and the second is always one, by the definition of the standard deviation itself. The third standardized moment is called skewness and measures the asymmetry of a distribution, while the fourth is called kurtosis and measures how heavy or light its tails are compared to a normal distribution. Because they are scale invariant, standardized moments make it possible to compare the shapes of distributions built from data measured in different units or at different scales.