Studies in Milk Secretion 97 



Where o-i is the standard deviation of the daughter's corre- 

 lated milk production, or butter-fat percentage and <x 2 the 

 standard deviation of the dam's milk production or butter-fat 

 percentage as the case may be. The calculation of the prob- 

 able error in the net change of the butter-fat can be easily made 

 from the above formula after the standard deviation of the net 

 butter-fat of the daughters and the standard deviation of the 

 net butter- fat of the dams are known. The formula for these 

 constants is approximately 



MjM. v, 2 + v 2 2 + 2r l2 vi v 2 (4) 



Where Mi equals the mean milk production, M 2 equals the 

 mean butter fat percentage, vj the coefficient of variation of 

 the milk production and v 2 the coefficient of variation of the 

 butter-fat percentage. 



Table 3 gives the probable errors derived by the above 

 formula. 



This table shows that little reliance can be placed in the 

 mean gain or loss in the milk production if the figures are based 

 on only two daughters and their dams, for the error due to 

 chance is likely to be 1 154.5 pounds or to be significant the dif- 

 ference of such pairs would have to be 3463.5, a difference only 

 rarely found. The error on as many as forty such pairs is not 

 very great (258.2 pounds). The probable error for the butter- 

 fat percentage on two pairs is so large that only differences 

 over 1.059 can De considered as more than indicative of a supe- 

 rior bull. For the bull that has as many as forty pairs much 

 smaller differences are required. The probable errors for the 

 number of pairs for the butter-fat are likewise large for those 

 sires which have only two pairs and relatively much smaller 

 for the sires with as many as forty pairs. 



In the balancing up of the true worth of a bull it is essen- 

 tial that these facts be taken into consideration in weighing the 

 evidence as to his merit. 



