WRIGHT: CORRELATION IN SUBGROUPS OF POPULATION 533 



with the number of individuals involved. Let <r x(t ), o- y(t ), r xy(t ) 

 be the values for the total population. The average standard 

 deviations within the subgroups can be calculated at once, as 

 the variability of the whole population is compounded of the 

 variability of the means of the subgroups and the independent 

 variability of the individuals about these means. 



Thus, <r x(t) = <r x ( m) + ot^y 



2 2 i 2 



°V(t) = °V(m) + Cy(g) 



It is evident that the mean values of x and y for the total 

 population are identical with those for the weighted means of the 

 subgroups. Take the intersection of these means as origin and 

 consider the contribution of a given subgroup to the term 

 sX ( t) F (t ) in the formula for the coefficient of correlation for the 

 total population. 



r 2X (t) F (t) 



r xy (t) 



n (t) °"x (t) Cy (t) 



Let X (m) , F (m) be the deviations of the means of the sub- 

 group from the origin. Let X (g) , F (g) be the deviations of any 

 point within the subgroup from the center of the latter. 

 X (m) + X (g) , F (m) + F (g) are the coordinates of the point. 

 For any such point there is a point of equal frequency at an 

 equal distance on the opposite side of the center of the sub- 

 group in the ideal case in which the subgroup is perfectly sym- 

 metrical about two perpendicular axes. Normal chance dis- 

 tributions tend to approach this ideal case. The coordinates of 

 this point are X (m) — X (g1 , F (m) — F (g) . 



The sum of the products of the coordinates of such symmetri- 

 cally placed points is as below. 



X( m ) F (m ) + X( m ) F (g ) + X( g ) F( m ) + X( g ) F( g ) 

 X( m ) F( m ) — X(m) 1(g) ~ X( g ) F( m ) + X( g ) F( g ) 



2X( m ) F(m) + 2X( g ) F 



(g) 



The sum of the products for all points in the subgroup, taken 

 thus in pairs, is n (g) X (m) F (m) + 2X (g) F (c) where n (g) is the 

 number in the subgroup. 



