148 



THE AMERICAN NATURALIST 



[Vol. LI 



Using Forsyth 's approximation, which i 



The gamma terms in (v) will, of course, be calculated 

 by some one of the well-known approximations (e. g., 

 Sterling's, Pearson's, Forsyth's) or by interpolation 

 from a table of factorials (PearP^). 



4. The proposal which I wish to make for the expres- 

 sion of a Mendelian result is that the expectation be 

 expressed as the quartile limits for each class frequency 

 in a second sample of the same size as the observed 

 sample. In using such an expression it must be clearly 

 understood that it does not measure the goodness of fit 

 of the distribution as a whole, because it takes no account 

 of correlations in errors. What it does give, in supple- 

 ment to the test, is the limits of probability of each class 

 frequency, taken by itself. 



The ordinary expression for a probable error {e. g., 

 P. E. mean = + .67449(o-/V»0) gives the quartile limits 

 {i. e., the limits within which one half the frequency oc- 

 curs) on the assumption that the distribution is Gaussian, 

 since in such a distribution of unit area the quartile limits 

 are .6744898 . . . times the standard deviation on either 

 side of the mean. But in our i)ersent work we are making 

 no assumption that the error distribution is Gaussian. 

 Consequently we must determine the quartiles directly 

 from the distribution. In cases where the number of 

 terms is not too great the ordinates may be calculated 

 from (iv) or (vi) and summed to find the quartile. In 

 many cases, however, this would be practically too tedious 



311 for High Values of n," Science, N. S., Vol. XLI, pp. 506-^507, 

 "On the Degree of Exactness of the Gamma Function Necessary in 

 Fitting," Ibid., Vol. XLII, pp. 833-834, 1915. 



