CORRELATION AS APPLIED TO FARM-SURVEY DATA. 3 



If, in two series of associated variables, as, say, the profit per 

 head and the weight per head in the data under consideration, there 

 is a tendency for a high value of the first to be associated with a 

 high value of the second, the variables are said to be correlated, and 

 the correlation is positive; while if a high value of the first is asso- 

 ciated with a low value of the second, and vice versa, the correlation 

 is said to be negative, and the best measure yet devised of the amount 

 of the correlation is the so-called coefficient of correlation. In Table 

 II is shown the calculation of the coefficient of correlation between 

 profit and weight per head. 



The method is as follows : 



1. Find the average value for each of the variables. Here the, average 

 profit per head is $0.78, and the average weight 834 pounds. 



2. Calculate the departure of the individual values frona the average. In 

 the case of record No. 1, the departure of the profit from the average is -f$11.29, 

 and of the weight, —49 pounds. 



3.- Find the square root of the average of the squares of these departures. 

 This is the so-called " standard deviation," and is a measure of dispersion or 

 the amount of variability of each variable. 



4, Find the algebraic sum of the products of each pair of individual depart- 

 ures, i. e., for each record, multiply the departure of the profit from the average 

 by the departui-e of the weight from the' average, and prefix the proper sign ; 

 then find the difference* between the sum of all the plus products and the 

 sum of all the minus products. 



5. Divide this result by the number of records and the standard deviation 

 of each of the variables in turn, prefix the proper sign, and tlie figui-e obtained 

 is the coefficient of correlation between the two factors under consideration. 



If there are approximately the same number of positive and nega- 

 tive products and they are of the same size, it will be evident that 

 there is no correlation, and this will be shown by the fact that the 

 coefficient of correlation will be zero, or nearly so. If high values 

 of the first variable are associated with high values of the second, 

 and low values of the first with low values of the second, most of 

 the products will be plus, and the greater their sum the closer will 

 be the correlation and the larger will be the coefficient obtained. 

 If a value of one variable below the average is generally associated 

 with a value of the other above the average, the correlation will 

 evidently be negative, and this will be shown by the fact that the 

 sum of the products will be negative, the degree of the correlation 

 and the size of the coefficient depending upon the size of this sum. 



Expressed algebraically, the coefficient of correlation, 



ncTj-ay 



(I) 



where T,xy is the sum of the products above mentioned, n is the num- 

 ber of pairs of variables (the same as the number of records) ; Ca? 



