What exactly are Moment 3 and Moment 4 in System.Math.MomentSkewKurtosis

Question

System.Math.MomentSkewKurtosis is declared as

procedure MomentSkewKurtosis(const Data: array of Double; var M1, M2, M3, M4, Skew, Kurtosis: Extended);

and returns M1, M2, M3, M4, Skew, Kurtosis where

M1 - Arthmetic Mean

M2 - Population Variance

Skew - Skewness

Kurtosis is Kurtosis

but what exactly M3 and M4 stand for.

The documentation in both Delphi Berlin http://docwiki.embarcadero.com/Libraries/Berlin/en/System.Math.MomentSkewKurtosis and Free Pascal http://www.freepascal.org/docs-html/rtl/math/momentskewkurtosis.html is vague.

Answer 1

Look at M3 and M4 (the 3rd and 4th order moment) as helper variables in order to calculate skew and kurtosis (extracted from sorce):

MomentSkewKurtosis: Calculates the core factors of statistical analysis: the first four moments plus the coefficients of skewness and kurtosis.

• M1 is the Mean.
• M2 is the Variance.
• Skew reflects symmetry of distribution: M3 / (M2**(3/2))
• Kurtosis reflects flatness of distribution: M4 / Sqr(M2)