INHERITANCE OE FECUNDITY IN DOMESTIC FOWL. 33I 



2. The evidence for a definite and clean-cut segregation of 

 high fecundity and lozv fecundity in gametogenesis is clear and 

 indubitable. The expected proportions of high producers and 

 low producers are closely realized in all the different types of 

 matings. 



3. Furthermore, the mean egg productions of the birds in 

 the several gametic classes are widely separated, showing that 

 the segregation is of perfectly distinct physiological entities. 

 Refined biometric tests are not necessary to show that the birds 

 carrying high fecundity hereditarily lay more than those with 

 low fecundity hereditary factors. The birds in the 'Over 30' 

 class have average winter productions from three to five times 

 greater than those of birds belonging to the 'Under 30' class. 



4. The agreement between observation and expectation for 

 the several types of mating is as close as could be expected 

 considering the nature of the material. The only discrepancy 

 of note is caused by the 10 birds with zero records, where none 

 are expected. In the detailed discussions in connection with 

 each mating it has been shown, however, that nearly all of these 

 10 cases, when studied individually, have a physiological expla- 

 nation, which makes it impossible to regard them as real excep- 

 tions to the gametic expectations. A determination might be 

 made of the 'goodness of fit' of theory to observation by Pear- 

 son's (42) method, were it not for the fact that that method 

 cannot be applied to cases like the present.''^ 



^^ The difficulty lies in the fact that Pearson's test depends upon a 

 variable 



{lUy — fn'r)''' ) 



where iii,- is the theoretical frequency and m'r the observed. Now ob- 

 viously in any distribution where one rriv is zero, the value of X: innst 

 be infinity, w'hatever may be the values of the other niv's or w'r's. That 

 is, if the theoretically expected frequency on any base element is nu- 

 merically zero the probability against the whole curve becomes infinite. 

 Thus, for example, suppose a system of frequencies like the following, 

 a type which is continually arising in Mendelian work. 



Class T 2 3 4 5 



Theoretically expected frequency. . . 595 827 68 o 06 



Actually observed freauency 594 828 67 t 96 



Now, it does not need a mathematical measure of any kind to tell 

 one that in this case the theoretical and actual distributions are in very 

 close aarreement. Yet, because the theoretical frequency on class 4 is 

 zero, the probability by Pearson's test is literally infinite against the 

 observed distribution being regarded as a random sample of a population 

 distributed in accordance with the theoretical frequencies. Pearson 



4 



