METHODS OF STATISTICAL ANALYSIS. 13 



Thus the sum of the products of the deviations of x and y from their 

 respective means for the whole series of individuals, divided by the 

 number of individuals considered, furnishes a mean product-deviation 

 which is a measure in absolute terms of the closeness of interdependence 

 of the two characters under investigation. 



To obtain a measure in relative terms (that is in a form to facilitate 

 comparison between unlike characters) some standard of the amount 

 of the deviation from the general means in the case of the two characters 

 is essential. The mean product-deviation must be expressed as a 

 fraction of the product of the deviations of the two characters in 

 the whole series of individuals from their respective means that is, 



Of ff X ffy. 



The measure of interdependence in relative terms is therefore 

 merely the ratio of the mean product-deviation discussed above to the 

 product of the two standard deviations in the whole series. Thus 



' xy 



is the measure of interdependence sought. 



For the illustration in hand, the athletes, we have numerically, 



3020.7188 3020.7188 n ftKC 

 r wh = - = = U.yoo 



12.867X245.12 3153.9590 



This is the familiar product-moment coefficient of correlation of the 

 statistician. 



The coefficient of correlation measures the closeness of interde- 

 pendence between two variables on a universally comparable scale, the 

 range of which is unity. Thus a coefficient of represents an absence 

 of all interdependence 5 between the two variables. A correlation 

 coefficient of 1 indicates perfect interdependence. Thus if there be no 

 correlation between x and y, the measurement of the x character 

 furnishes no information whatever concerning the magnitude of the y 

 character in the same individual. If, on the other hand, there be perfect 

 correlation a practically unknown quantity in biological work the 

 magnitude of the y character is known as soon as the x character has 

 been measured. 



Empirically, the correlation coefficient is generally found to be 

 positive in sign, but it may be either positive or negative. When y 

 becomes larger as x increases in magnitude the correlation is positive 

 in sign. When y decreases as x increases, correlation is negative in 

 sign. The correlation formula is so written that the sign is automatic- 

 ally given in the process of determining the constant. 



s There are conditions under which this is not true, but for the purposes of this volume the 

 statement is practically valid. 



