CORRELATION 



455 



of cholera subjects to black pigs is higher than the ratio to pigs 

 in general, then we should conclude that an actual positive corre- 

 lation exists between the black color and this particular disease. 

 On the other hand, if this ratio should be below the ratio of 

 pigs in general, then we should conclude that black pigs are 

 less susceptible to this disease than are pigs of other colors, and 

 that a negative correlation exists, assuming always equal oppor- 

 tunities for infection. This is the only correct method of study, 

 and it would not be safe to conclude that black pigs are pecul- 

 iarly susceptible simply because most of the pigs that died under 

 our observation happened to be black, for black pigs are more 

 numerous in the cholera belt than all others combined, and 

 under probability alone their absolute mortality must be higher. 

 When expressed in the form of a ratio, however, the truth comes 

 to the surface. 1 



Similarly we may ask the question whether different species 

 of plants or animals tend to attract or repel each other when 

 thrown together in the same territory. Here again the first 

 step is to find the ratio of association under free operation of 

 independent probability, a ratio based on the relative numbers 

 of individuals of the species in question and the extent of terri- 

 tory, first where no opportunity for association is possible, and 

 second, where such association is possible. If the two ratios 

 differ, then we infer that some degree of correlation exists. 



SECTION II CALCULATION OF COEFFICIENTS OF 

 CORRELATION 



When the presence or absence of the characters in question 

 is absolute, as red or black hair, presence or absence of horns, 

 then the correlation is expressed by a single ratio, as we have 

 seen. But most cases are not of this extreme simplicity ; for 

 example, it is said that white cats are deaf. If now all white 

 cats are deaf, then the correlation between albinism and loss of 

 hearing power is absolute, and is expressed by the coefficient i. 



1 Perhaps it ought to be remarked that this illustration is taken purely at 

 random, as no studies have been made as to the relation between color and this 

 particular disease. 



