a 2 



= 1 





m a 



a 3 







s ; 





«r 



a 4 



m 4 





s 





X 



The first four standardized moments are 



a-L = (12a) 



(12b) 

 (12c) 



(I2d) 



The third moment, a 3 , is a measure of the skewness of the 

 distribution. A positive value indicates a distribution with 

 a longer positive tail than a negative tail. 



The fourth standardized moment, a 4 , is a measure 

 of the kurtosis of the distribution. In some cases it. is a 

 measure of the "peakedness" of the distribution, though it 

 is now understood that the length and size of the tails are 

 very important in this measurement. 



For a normal curve the values of a 3 and a 4 will be 

 and 3, respectively. We redefine the skewness and kur- 

 tosis as 



Q, r « 3 (13a) 



g 2 = « 4 - s (i3b) 



so that g is for a normal curve. 



It is not very likely that the third and fourth moments 

 of a random sample will be zero. Depending on the distri- 

 bution and on the actual sample values, the third and fourth 

 moments will have some value different from zero. To de- 

 termine whether this difference is significant, it is neces- 

 sary to use the variances of the third and fourth moments. 4 



varfo, ) = 6N(N-l)(N-2)- 1 (N-l)- 1 (N-3)- 1 (14a) 



20 



