6 BULLETIN 504, U. S. DEPARTMENT OF AGRICULTURE. 
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 w, the value per hundredweight v, and the date of 
sale d, the gross correlation coefficients are: 1 r wu =+.56; r vd =-\-.61; 
r wd =-\-.60. Applying the formula (III), we have: 
+ .56-( + .61)( + .60) _ J _ 
wv ' d A /(l--61 2 )(l-60 2 ) + 
This value. +.31. is appreciably smaller than the value, +.56, 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. 2 
In the general case for six variables, the total number considered 
in this paper — 
fgb.cde ^af.cde * ^bf.cde (T\f\ 
a "'"V(i-^.*)(i-'V.*) ' (} 
Tab'cdef is the net coefficient of correlation between a and 6, when the 
four factors, c, d, e, and /, are taken into account: r a& . cde , r a f. C de, and 
r^f.cde are the coefficients of correlation between the two variables 
before the period in each case when e, cU and e are taken into account. 
COMPUTATION OF THE COEFFICIENTS. 
The first step in the arithmetic was 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 coefficients. 
2 Yule, G. U. : " Introduction to the Theory of Statistics,*' p. 229 et seq. 
