94 PROCEEDINGS OF THE AMERICAN ACADEMY. 



difference, divide the deviation of the subjects by their index of variabil- 

 ity and the deviation of the corresponding relatives by their index of 

 variability. If now, the correlation is perfect, those causes which have 

 produced a deviation from the mean in the right leg will act in precisely 

 the same degree on the left leg also, and thus the deviation of any rela- 

 tive will not differ from the deviation of the corresponding subject. If, 

 under these circumstances, we divide the mean deviation of the relatives 

 by that of the subjects, the quotients will average 1. This average 

 quotient is called the Index of Correlation. Thus, the index of perfect 

 correlation is 1. 



Let us suppose, on the contrary, that there is no correlation whatever 

 between the number of glands in any subject and in the corresponding 

 relative, then, no matter what the number of glands in any subject, the 

 number in the corresponding relative is just as apt to be large as small, 

 and will be equal to the average number of glands in the whole group ; 

 in other words, no matter what the deviation of the subject is, that of the 

 relative will be 0. The average quotient obtained, under these cir- 

 cumstances, by dividing the deviation of relatives by the deviation of the 

 subjects, will consequently always be 0. Thus the index of entire lack 

 of correlation is 0. 



An inverse correlation, in which a positive deviation of the subject 

 from the mean shall always be accompanied by a negative deviation of 

 the relative, will be represented by a minus quantity. Thus the correla- 

 tion of any two sets of compared organs will lie between +1 and — 1. 

 The size of the fractions lying between ± 1 and will serve to indicate 

 the degree of correlation. 



The quotient, r, obtained by dividing the deviation (always in units of 

 the average deviation) of the left legs by that of the corresponding right 

 will not be the same for all the lots of individuals. The true index of 

 correlation, R, will be found by taking the average of all the ratios, 

 r, /, /', /", etc. This process of finding R may be somewhat abbre- 

 viated from the following considerations. We have seen that 



Deviat. of Rel. d^ 



also that 



_ Avg. Dept. of Rel. A^ . 



Deviat. of Subj. dg 



Avg. Dept. of Subj. Ag 



R^- (r + r' + r" + . . . . r") = - 2r ; 



