INHERITANCE OF FECUNDITY 203 



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

 fecundity and low fecundity in gametogenesis is clear and indubit- 

 able. The expected proportions of high producers and low pro- 

 ducers 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 aver- 

 age 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 con- 

 sidering 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 ekch 

 mating it has been shown, however, that nearly all of these 10 

 cases, when studied individually, have a physiological explana- 

 tion, which makes it impossible to regard them as real exceptions 

 to the gametic expectations. A determination might be made of 

 the 'goodness of fit' of theory to observation by Pearson's (42) 

 method, were it not for the fact that that method cannot be 

 applied to cases like the present. i- 



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



[ mr ) 



where uij- is the theoretical frequency and m'l- the observed. Now obviously in 

 any distribution where even one niT is zero, the value of X2 must be infinity, what- 

 ever may be the values of the other wir's or m'p's. That is, if the theoretically 

 expected frequency on any base element is numerically zero the probability against 

 the whole curve becomes infinite. Thus, for example, suppose a system of fre- 

 quencies like the following, a type which is continually arising in Mendelian work. 



Class 



Theoretically expected frequency.. . 

 Actually observed frequency 



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 agreement. 



