THE PROBABLE ERROR OF A MENDELIAN 

 CLASS FREQUENCY^ 



DR. RAYMOND PEARL 



1. With the increasing volume of Mendelian experi- 

 mentation there is an ever-growing need for adequate 

 and clearly understood tests for the statistical significance 

 of differences between observed results and expectation. 

 A number of different methods of making such tests have 

 been proposed and used by different workers. For ex- 

 ample, early in the discussion of Mendelism, Weldon^ 

 made use of the ordinary, and frequently inadequate, ex- 

 pressi on for the standard deviation of a sub-frequency 

 a =z ^/npq. Johannsen^ has also made much use of this 

 method. It has several defects. In the first place it as- 

 sumes the Gaussian distribution of the errors, an assump- 

 tion not often strictly warranted, as Pearson^ has clearly 

 shown, and in many cases grossly in error. In the second 

 place it is not even approximately adequate under certain 

 extreme conditions (frequently realized in Mendelian 

 work) of class frequency. Harris^ has proposed the 

 ''goodness of fit" test for comparing observed and ex- 

 pected Mendelian distributions. There are several fea- 

 tures of this method which greatly limit it for such use. 

 Among these are (1) its failure to make correct allowance 

 for ''tail" frequencies (it is just this class of very small 

 frequencies which one most often wants to test in prac- 



1 Papers from the Biological Laboratory of the Maine Agricultural Ex- 

 periment Station No. 108. The author is greatly indebted to his assistant, 

 Mr. John Eice Miner, for the laborious arithmetic involved in Section 6 of 

 the paper. 



2 Weldon, W. F. R., "Mendel's Laws of Alternative Inheritance in Peas," 

 Biometrika, Vol. I, pp. 228-265, 1902. 



3 Johannsen, W., "Elemente der exakten Erblichkeitslehre," Zweite Aus- 

 gabe, Jena, 1913. 



4 Pearson, K., ' * On the Influence of Past Experience on Future Expecta- 

 tion," Fhil. Mag., March, 1907, pp. 365-378. 



B Harris, J. A., "A Simple Test of the Goodness of Fit of Mendelian 

 Ratios," Amer. Nat., Vol. XLVI, pp. 741-745, 1912. Cf. also Pearson, 

 K., and Heron, D., ' ' On Theories of Association, ' ' Biometrika, Vol. IX, pp. 



159-315, 1913. 



