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THE AMEBIC AN NATURALIST [Vol. XLVI 



"probable error" of the number of individuals of any class, 

 say p, is 



#p = . 67449 \hipq. 

 Now while Professor Weldon's use of this formula for the 

 simple 3 : 1 ratios seems quite proper, the same can not be said 

 for Professor Johannsen's generalization. This is true for three 



(a) The formula is valid only when neither n, p nor q is 

 small. In polyhybrid ratios p or q may be relatively small. 4 

 It is then quite idle to use the probable error suggested, unless 

 n be large, which unfortunately is generally not the case. 



(b) Even when p is not so low as to render the use of the 

 conventional formula for the probable error open to question, 

 it is very laborious to calculate the probable errors for the fre- 

 quency of each class. 5 



(c) It is not only cumbersome and laborious, but theoretically 

 unjustified to test the validity of a given ratio by the determina- 

 tion of the probable error of one or of all of its individual com- 

 ponent groups. The random deviations of the class frequencies 

 are not independent, but correlated. "We must have a usable 

 criterion of the goodness of fit of the theory to the data as a 

 whole. 



Such a criterion was furnished several years ago by Pearson. 6 

 Its applicability to the problem of testing the goodness of fit of 

 Mendelian ratios seems obvious, but since, as far as I can ascer- 

 tain, it has nowhere been applied to this problem, it seems 

 worth while to call the attention of students of genetics to its 

 usefulness. 



x ^S{(o-cr/c}, 

 where o is observed frequency of any class, c is calculated fre- 

 quency on the basis of Mendelian theory and S indicates a 

 summation for the several classes distinguishable in the ratio 

 under consideration. 



P, a measure on the scale of to 1 of the probability that 

 * For example, Johannsen (loc. cit., p. 405) tables values for p = 3/4, 

 q = 1 /4 to p = 63/64, q = 1/64. 



•Pearson, K., "On the Criterion that a Given System of deviations 

 from the Probable in the Case of a Correlated System of Variables is Such 

 that it Can be Reasonably Supposed to have Arisen from Random Sampling," 

 Phil. Mag., 50: 157-175, 1900. 



