66 GENETICS AND EUGENICS 



been entered in the table, we may proceed to calculate ^ a 

 coefficient of correlation which will be a measure of the 

 extent to which men vary in weight as they vary in height. 

 Its numerical value will lie between and 1. 



It is evident that the correlation would be most complete if 

 men invariably increased in weight as they increase in height. 

 The entries in the table would then be distributed in a single 

 diagonal row running across the table from its upper left-hand 

 corner to its lower right-hand corner. We should infer that 

 in such a case the two completely correlated phenomena were 

 due to the same causes or contingencies exactly. Our numer- 

 ical coefficient of correlation would in such a case be -|- 1. 



In reality such correlation as this rarely, if ever, occurs in 

 biological material. We know that men of the same height 

 vary in weight and vice versa. For weight does not depend 

 upon height alone but also upon width and thickness and 

 specific gravity. It does however depend sometvhat upon 

 height, and so our table would show incomplete correlation, 

 which would be expressed by a coefficient less than 1 but 

 greater than 0. 



^ The coefficient of correlation is calculated by the formula 



r = . 



n <Tx <^y 



in which r is the coefficient of correlation, Dr and Dy are the deviations of each 

 observed group of individuals from the respective means of height and weight, S sig- 

 nifies that the sum of the products indicated is to be taken, n is the total num- 

 ber of individuals observed, and ai and ay are the standard deviations for height 

 and weight respectively. To express in the form of a rule the procedure to be 

 followed in calculating the coefficient of correlation between (say) height and 

 weight: First find the average height and the average weight of all individuals ob- 

 served, then their standard deviation in height and their standard deviation in 

 weight. Next determine for each square of the table its deviation from the aver- 

 age height and average weight respectively. Find the product of these two devia- 

 tions (regarding signs) and multiply it by the number of individuals recorded 

 in the square imder consideration. After such a product as this has been found 

 for every square in the table, the products are to be added (regarding signs) and 

 this sum is to be divided by the product of the two standard deviations times the 

 total number of individuals observed. There are several short-cuts by which the 

 calculation as here described may be shortened or simplified. For a description of 

 these the reader is referred to the special works of C B. Davenport (1904), Eugene 

 Davenport (1907), and Yule (1912). 



