330 J- ARTHUR HARRIS 



The first and second moment of y for each grade of x, i.e., 2(;y x ) 

 and ^(y/), are easily determined. The first of these will have been 

 obtained in computing the population product moment, S(xy), by the 

 method cited above. 



Remembering that 2(^) and 2(^ 2 ) are the first and second moments 

 about zero as origin for the several y arrays of x, the moments of z 

 for each array are 



%M = 2 (3O -- n t px 



From these the means of arrays which are required for plotting the re 

 gression curve are given at once by 



**= [2(y x ) -- n f px~\/n x 

 For the whole population the first moment of 5 about o as origin is 



n f x = o 

 The second moments for the individual arras are 



and for the population 



SS,(s t *) == S\ 2(3 . c 2 ) - - 2 

 or in some cases more conveniently for actual computation, 



where ^ denotes summation of values of y arrays of x, or throughout 

 the population. Thus 



t is quite possible to determine the correlation between x and s, the 

 deviation of y from its probable value, directly. 



Remembering that [S(^) -- n ,/&amp;gt;.r] is the first moment about zero as 

 origin of z for any array, the product moment for the population is 



where S denotes summation of the values of the 3 arrays of x. 



Practically it is more convenient to determine the product moment 

 from 



where S(xy) and S(S) are the product moments of x and y and the 

 second moment of x for the population. 



