CORRELATION 



457 



correlation is positive. Whenever ad and be become equal 



the formula becomes - - = o, or no correlation ; whenever 

 ad + be 



b or c becomes zero, then the formula becomes -- = i or 



ad 



perfect positive correlation ; and whenever a or d becomes 



zero, then the formula becomes = i , or perfect negative 



correlation. 



This is the simplest formula proposed that will meet the 

 necessary conditions of the case. Pearson l has proposed several 

 others that are much more complicated, and that differ slightly 

 as to results. Strange as it may seem, the problem is a com- 

 paratively new one, though the question involved is fundamental 

 and very old. Though other methods are in use for special 

 cases we may safely use Yule's formula for all ordinary cases 

 of association where the question is simply as to presence or 

 absence, without involving considerations of degree ; that is to 

 say, when the question is whether or not the cats are deaf, 

 without reference to degrees of deafness ; whether or not the 

 patient has smallpox, without reference to the severity of the 

 attack. 



When, however, the question is one of possible correlation 

 between characters present in varying degrees, as size, weight, 

 amount of milk, etc., the problem would seem at first thought 

 to be far more difficult ; but in truth it has been much more com- 

 pletely worked out than the preceding question. 



For example, what is the correlation between length and cir- 

 cumference in ears of corn ? In general, long ears are also large 

 ears, but many can be found that are long and slender, many 

 that are short and small, and still others that are short and large. 

 In other words, the two characters, length and circumference, 

 are so related that the two maxima may appear together, the 

 two minima together, the maximum length and the minimum 

 circumference and vice versa, and all grades between. What 

 now is the correlation ? To answer a question thus complicated 

 we construct what is called a correlation table. 



1 Philosophical Transactions of the Royal Society, CXCV, 1-47. 



