6 BULLETI:N' 504, U. S. DEPAEiMENT OF AGEICULTURE. 



What would have been the coefficient of correlation between weight 

 and price if all the calves had been sold on the same date ? Calling 

 the weight u', the value per hundredweight v, and the date of 

 sale d, the gross correlation coefficients are:^ ^^^=+.56; rvd=4-'61; 

 r^cd=+.60. Applying the formula (III), we have: 



+ .56-( + .61)( + .60) _ 



This value, ^-^l? is appreciably smaller than the value, +-5^; of 

 the gross coefficient, showing that the apparent correlation between 

 weight and price is partly, but not entirely, due to their mutual 

 correlation with the date of sale. 



This theory can be applied to the case of several variables by a 

 simple extension of the formula.^ 



In the general case for six variables, the total number considered 

 in this paper — 



^ab.cde ^a f.cde ' ^bf.cde ^ /■TV> 



■\ \^ — ^ ai .cde) K^ — ^ bf .cde) 



ifab-cdef is the net coefficient of correlation between a and &, when the 

 four factors, <?, d^ e, and /, are taken into account : rah-caet ^af-cdc, and 

 ^bf-cde are the coefficients of correlation between the two variables 

 before the period in each case when c, d, and e are taken into account. 



COMPUTATION OF THE COEFFICIENTS. 



The first step in the arithmetic w^as the computation of the gross 

 correlation coefficients. As stated above, the variables or factors 

 considered were: (1) The profit or loss per head; (2) weight; 

 (3) value per hundredweight; (4) total value of feed consumed per 

 head; (5) cost per head at weaning time; and (6) date of sale. 

 These six variables, if taken two at a time, can be combined in 15 

 different ways. The first calculation was to find the coefficients of 

 correlation between these 15 different pairs. In Table III these are 

 the first values given. The effect of every other factor on these 

 gross coefficients was then eliminated by successive applications of 

 formulae III and IV. As an example, take profit and weight, 

 the first pair of variables correlated. The gross coefficient was first 

 corrected for the effect of value per hundredweight, value of feed 

 consumed, initial cost, and date of sale, in turn. Then the effect 

 of these four factors was considered, taking them two at a time. 

 That is, the correlation was determined when both the value per 

 hundredweight and the cost of feed were taken into consideration 

 at the same time. When the effect of all these factors, taking them 



1 See Table III : Correlation coeflBcients. 



« Yule, G. U. : " Introduction to the Theory of Statistics," p. 229 et seq. 



