PARTIAL CORRELATIONS. 



37 



Now suppose that we so select our group of individuals that they shall 

 all be exactly the same with reference to the character a;,, or, in other words, 

 eo that after selection there shall be no variation in respect to x r The 

 standard deviation after selection, s,, will thus of course be zero. Putting 

 s, =0 in (i) we have at once 



**,= 



which is the well-known expression for a partial correlation coefficient. 

 This coefficient measures the correlation between x 2 and x 3 in a group of 

 individuals where x 1 is constant. If, for example, we let x l denote length 

 of cephalothorax, x 2 length of the great chela, and x 3 length of the carpo- 

 podite of the cheliped, then t. 23 measures the correlation between the last 

 two characters in a group of individuals all having the same length of 

 cephalothorax. 



The partial correlations of the different joints of the legs with each 

 other were studied in order to get further light on the factors which 

 influence the degree of the gross correlations. It was decided to deter- 

 mine the partial correlation between every possible pair of joints avail- 

 able in our data when the cephalothorax length was made a constant. 

 The length of the cephalothorax may be taken as an adequate index of 

 the size of the body, and was on this account chosen as the character to 

 make constant in the calculations. Calculating from (n) above, and 

 making cephalothorax length in every case the x l character, we have found 

 the system of partial correlation coefficients between the different joints 

 of the legs given in table 18. In the calculations the gross coefficients 

 which were substituted in equation n were kept to six places of figures. 



TABLE 18. Partial correlation coefficients between the joints of the legs, the cephalo- 

 thorax length being kept constant. 



