2G8 PROP. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
correlation in species, but the equally interesting point of the extent and manner of 
its variation in local races. 
(5.) Regression, Uniparental Inheritance, and Assortative Mating, 
(a.) General Formula. —On the basis of the above discussion we can obtain the 
formulae requisite for calculating scientific measures of uniparental inheritance 
and assortative mating. 
Let male or female parents solely be kept in view, and let male or female parents 
be considered which have an organ or measurable characteristic differing h from that 
of the general population of male or female parents. Then the frequency of a 
variation x in the same or any other organ of the offspring is given Iry 
N 1 _AL - w it le r 
2= — — — e *W(1 -r 2 ) °YT 2 ( 1 —r*) 
Zira^c, x /(l — r~) 
The offspring, therefore, have variation following a normal distribution about the 
mean 
O'! 7 
x 0 = r — h, 
G 2 
and with standard deviation oq ■v/0 - r>). 
Hence, by our definition, the coefficient of regression = xjh = roq oq, and the 
variability of the offspring of the selected parents is reduced from that of the 
general population of offspring in the ratio of ^/(l — r~ ) to 1. We thus have a 
measure of the manner in which selection of parents reduces the variability in 
offspring, i.e., tends to make the latter closer to a definite type. This result is achieved 
even with promiscuity in the case of one parent, if there be selection in the case of 
the other. The greater closeness of approach to type when both parents are selected 
will be dealt with under biparental inheritance. 
We note that the coefficient of regression and the restriction of variability are the 
same whatever type of parent be adopted, or the closeness with which selection leads 
to a given type of offspring is independent of the parent adopted and the type of 
offspring which results from this parent.* 
* This is, of course, true of the regression and variability of the array corresponding to any type 
whatever, when frequency follows the normal law. Mr. G. U. Yule points out to me that if the 
coefficient of regression be constant for the arrays of Ml types, then it follows that whatever be the law of 
frequency, the coefficient of regression must = roj/o-g, where r— S (xy') / (na-^a-f). This much generalises 
the formula. At the same time, in the case of skew-correlation, the coefficient of regression usually 
varies with the type, and the fundamental problem is to determine what function it is of the type. Let 
bridegrooms of age differing by p years from the mean age of all bridegrooms have an array of brides 
with a mean age differing q years from the mean age of all brides ; then fjq is not constant for all 
values of p. 
