is defined as the mean value of x, m ' - x. 



Another more important set of moments is obtained 

 by changing the origin to the arithmetic mean. Equation 8 

 defines the moments about the mean. 



m = £ p.(x) (x. - x\ (8) 



For computing purposes, the relations between the 



m and the Tn ' are convenient. Expressing the m in terms 



of the rn ' we have the relations 

 r 



m 1 = m 1 ' (9a) 



m 2 = m 3 ' - (ot 1 'f (9b) 



m 3 = m 3 ' - 3m a 'm 1 ' + 2(% ') 3 (9c) 



ff2 4 = ot 4 ' - 4m 3 , m 1 ' + Qm 3 '{m x 'f - 3(m ± ') 4 (9d) 



Grouping errors are negligible, so Sheppard's 

 corrections are not applied. 



- These moments can be expressed in standard units 

 by the use of a standardized variable z, by dividing the 



variable x by s , the standard deviation. 



J x 



(x-x) 

 z - 



s 



X 



(10) 



The standardized moments are defined by the equations 



772 



a = — — , for r = 1, 2, 3, and 4 (11) 



r r 



s 

 x 



19 



