464 TRANSMISSION 



For the array corresponding to. 5 inches, 



- 7.7 X 2 



-6.7 x 4 



This gives 297.6. 



- 2.8 x - 5.7 x 7 

 -4.7 x 2 

 -3-7 x 4 



Treat all arrays in a similar manner, and, finally, divide the 

 sum of all the products thus obtained (that is 4947.2) by the 

 product of the two standard deviations and the number of 

 variates, being careful always to preserve the full distinction as 

 to plus and minus signs. This gives 



r = ^ = 0.87, the correlation coefficient. 



993 (i-5 7) (3-63) 



The mathematical derivation of this coefficient as a measure 

 of correlation involves too much mathematics to be given here. 

 It may be noticed from the common-sense standpoint, however, 

 that it seems to be a good measure of correlation. To appre- 

 ciate the meaning of this coefficient, it should be recalled that 

 we take the products of both deviations for every individual in 

 the table, add these products, and divide the result by the 

 number of individuals. This gives the average of all the 

 products of both deviations. We then divide this average 

 product of the individual deviations by the product of the two 

 standard deviations, thus securing an expression whose value is 

 influenced by the deviation of both characters with reference 

 each to the other. 



It requires but little mathematical insight to see that if the 

 correlation is positive and considerable, positive values of the 

 two characters correspond and negative values correspond ; and 

 further, that all the products of deviations are positive. This 

 makes for a large correlation coefficient. On the other hand, if 

 no correlation exists, for any value of one character we may 

 expect in the long run equal and opposite deviations of the 

 other character, which makes the sum of products of deviations 

 very small. This common-sense examination indicates the real 

 nature of the correlation coefficient. 



Fourth step. Find the probable errors in the determined values. 

 Those in the means and standard deviations are computed by 



