374 



THE AMERICAX NATURALIST [Vol. LI 



from the usual. This seems to indicate that the old method is 

 wholly inadequate, but further examination shows that the differ- 

 ence is not due so much to method as to the fact that Pearl has 

 calculated something with a different significance from the usual 

 probable error. A cross of Mendelian heterozygotes (Blue Anda- 

 lusian fowls) gave three classes of young in the numbers 

 14 : 33 : 11. Expectation is 14.5 : 29 : 14.5. Pearl assumes that a 

 first sample of 58 has given exactly expectation and then cal- 

 culates the quartile deviations for each class in a second sample 

 of 58. The results are given as 3.13 for the heterozygous classes, 

 3.55 for the homozygotes which indicate an excellent fit of ob- 

 servation to expectation. By the usual method, if a first sample 

 of 58 had given exactly 14.5 black chicks and nothing were known 

 of any theoretical expectation, the probable error in a second 

 sample of 58 is measured by the probable error of differences. 

 The probable error of either sample as given by the formula 

 .61i5\/npq is 2.22. The probable error of differences by the 

 usual formula .6745V o-i^ + o-j- is 3.15. This does not differ ap- 

 preciably from Pearl's quartile of 3.13. Neither of these 

 methods, however, gives what we really wish to know, the close- 

 ness of fit to Mendelian expectation. We have a theoretical ex- 

 pectation which is not based merely on a particular sample of 58, 

 but which should hold with increasing accuracy the larger the 

 first sample taken. With an infinite first sample, the formula 

 given by Pearl reduces to the usual one, .6745 Vwpg giving a 

 quartile of 2.22. This is less lenient to the discrepancy between 

 expectation and observation than the first result, but the fit is 

 still not bad. In a second illustration which is given, we do 

 have two samples and no theoretical expectation suggested. The 

 usual method of comparing samples of different sizes would be 

 to find the standard deviation of differences on a percentage 

 basis. The ]»orrejitajie standard deviation for a sample of n 

 individuals is \ pq for a sample of m individuals is \/p'q'/m 

 and for (iitrm.n.M's is V (pq/n) + (p'q'/m). The expected 

 staudaid deviation of a sample of m individuals is, however, 

 m\/ {pq/n ) + {pq, m) if p and q are based merely on the first 

 sample as in Pearl's illustration. The formula given by Pearl 

 for the standard deviation rapidly approaches this form for large 

 values of m and n. Following are the results given by the long 

 method, by an approximation given by Pearl and by the usual 

 one just cited. 



