264 Prof. K. Pearson. On a Criterion which may [Mar. 4, 



on a consideration of the variability of the offspring due to parents of 

 given types. Luckily the three theories give us totally different values 

 of the variability of an array of offspring due to parents of given 

 types, and we have in this question of variability a crucial test of the 

 applicability of one or other of the theories to the inheritance of a 

 given character. 



In order to bring this point out I must briefly consider the 

 variability of arrays of offspring under the three theories. 



(2) Variability of an array of offspring on the pure statistical theory 

 developed as the "Law of Ancestral Heredity." 



If o- c be the standard deviation of the offspring, say of one sex, pf c 

 their correlation with father, p mc with mother and p/ m the coefficient of 

 assortative mating, then 



2 = o- a / ~ Pf ° 2 ~ Pm ° 2 - pfm2 + %PfcPmcpfm \ 

 V V 1 - Pfm 2 J 



is, whatever be the nature of the frequency distribution provided the 

 regression be linear, the mean of the standard deviations of all the 

 arrays due to parents of given types. If the characters be distributed 

 according to the normal law of deviation, then 2 will not only be the 

 mean of all the array standard deviations, but the actual standard 

 deviation of each array. If, therefore, the character selected be in each 

 generation distributed according to the normal law, we should expect to 

 find that if we take all pairs of given types, the offspring due to such 

 pairs will have a variability given with reasonable closeness by the above 

 result. If we deal with all the offspring due to fathers, say, of a given 

 type, the mean standard deviation of the arrays will be o- c ^/(l - p/ c 2 ), 

 and in so far as the distributions are approximately normal the standard 

 deviations of all arrays will be the same.* Hence arises the importance, 

 when we use 2 as the variability of the offspring, of showing that the 

 regression for the given character is linear, and that the frequency is not 

 widely divergent from a normal distribution. These points were dealt 

 with by Mr. Gal ton in his very first investigation of the subject. He 

 actually considered in the case of stature whether some of the offspring 

 followed the father and some the mother, and showed that 2 did not 

 vary sensibly from array to array. f Subject, therefore, to a demon- 

 stration for each character that the frequency is approximately normal 

 and the regression linear, we see that the purely statistical theory of 



deviations from type (' Roy. Soc. Proc.,' vol. 62, p. 387), and the offspring type 

 itself may be wholly different from both parental types, exceeding or falling short 

 of them. 



* The property that 2 is the mean of the standard deviation of all the arrays 

 was first stated by Yule, ' Eoy. Soc. Proc.,' vol. 60, p. 477, for the case of linear 

 regression. 



f ' Natural Inheritance,' pp. 89 — 90, and Table 10, p. 207. This investigation 

 seems to have escaped Dr. Boas, see loc. cit., p. 530. 



