114 
Heredity in Production of Butter Fat 
Neglecting other forms of selection than that of meeting requirements for 
entry, 
s = Si, a = ai, fjb = f^i. 
Then from (1) 
_ -<a- + Vya^ + 4i^-^(l-/^-) 
' ~ 2R(l-fi') ^ ^' 
The value of s for cows of various ages is known from Table I., but it is 
necessary to evaluate cr in order to obtain fi. To determine a we make use of the 
fact that the frequency distributions which give s are truncated. The requisite 
theory for obtaining the standard deviation of a normal distribution from a trun- 
cated portion of it has been given by Pearson*. 
Let N, 11 be the number in the total group and in the " tail " respectively, h 
the distance from the mean of the total distribution at which the distribution is 
truncated, the deviation of the mean of the selected group from the minimal 
requirement, 2 the standard deviation of the tail about its own mean, 
J w 
where Ji = - . Then 
iV' 
^;^,=ir^r (3) 
are the formulae to be used for our purpose. Further relations are given for the 
determination of N from the tail, but, bearing on the application of these formulae, 
it is pointed out that the effect of random sampling on the frequency at the stump 
of the tail is very influential in modifying the results. It is obvious that, in our 
problem, the frequencies near the stump are likely to be less than what we should 
obtain from the truncated portion of a random sample of the total group, for the 
reason that a cow is likely to be left out when she can barely meet the require- 
ments. Hence, instead of trying to determine 7i/N from the " tail," we determine 
this ratio by direct observation, as we have the means of making a fair estimate 
of the number of cows N in milk from which the n are selected. The basis of the 
method is found in the fact that the Holstein-Friesian Association maintain a 
Herd Book in which animals are registered, in general, when they are calves. 
Substantially all the female calves that are born are registered by such owners 
as use the Advanced Registry. By thus going to the Herd Book, we find the 
number of the female calves registered together with the age of calf and the Herd 
Book number of the dam, so that we can find the age of the dam. As a cow is in 
milk for each calf registered, we double the number of female calves registered to 
obtain an estimate of the number of cows in milk. We have excluded a few 
owners who buy and sell extensively, as the transfer of animals might prove a 
source of error. We have searched the records for the year May 15, 1906 to 
May 15, 1907, and the results may be exhibited in the following form : 
* Loc. cit. pp. G3— 68. 
