46 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
is selective pairing of dam and sire. We will suppose grandsire, dam, and sire to be 
above the average, and investigate what proportion of the produce will be above the 
average. As numbers very like those actually occurring in the case of dogs, horses, 
and even men, we may take 
Correlation of grandsire and offspring . = '25 
,, sire or dam and offspring = '5 in both cases 
,, sire and grandsire. . . = '5 
Selective mating for sire and dam. . . = '2 
We will suppose zero correlation between paternal grandsire and dam, although 
with selective mating this may actually exist. * We have then the following- 
system :— 
H.4 — 25, ^ 24 , — 5, 
34 
= ‘ 0 , 
23 
— •9 
12 
= A 
13 
0 . 
Hence, substituting these values in (lxxxvii.), we find—after some arithmetic : 
(Q - Qo)/Qo = 1A851. 
But Q 0 is the chance of produce above the average if there were no heredity 
between grandsire, sire, and dam, and no assortative mating. 
Hence it equals \ X \ X w X = — .'. Q = '1553 N. 
Or, of the produce '5 N above the average, '1553 N instead of '0625 N are born of 
the superior stock owing to inheritance, &c. In other words, out of the '5 N above 
the average, '1553 N are produced by the stock in sire, dam, and grandsire above the 
average, or by '1827 of the total stock, t The remaining '8173 only produce '3447 N, 
or the superior stock produces produce above the average at over twice the rate of the 
inferior stock. Absolutely, the inferior stock being seven times as numerous produces 
about seven-tenths of the superior offspring. 
Illustration IX. Effect of Exceptional Parentage .—Chance of an exceptional 
man being born of exceptional parents. 
Let us enlarge the example in Illustration II., and seek the proportion of exceptional 
men, defined as one in twenty, born of exceptional parents in a community with 
assortative mating. 
* A correlation, if there be substantial selective mating, may exist between a man and his mother-in- 
law. Its rumoured absence, if established scientifically, would not, however, prove the non-existence of 
selective mating, for A may be correlated with B and C, but these not correlated with each other. 
t The proportion of pairs of parents associated with a grandsire above the average was found by 
putting -5, -2, and 0 for the three correlation coefficients in (lxxxv.). In comparing with Illustration II., 
the reader must remember we there dealt with an exceptional father, 1 in 20, here only with relatives 
above the average—a very less stringent selection. 
