1919] Frost: Mutation in Matthiola 147 



For example, table 27 gives the percentage of large-leaved plants 

 among the 357 progeny of the 20 large-leaved parents as 49.0 ± 1.8 



per cent. This probable error is given bv .6744898 "V— , where 



n 



p^49.0 per cent, g=:51.0 per cent, and n= 357. These 357 

 progeny, as table 39 indicates, came from 20 parents which contributed 

 an average of 17.85 progeny each, and the actual standard deviation 

 of the percentage in these 20 sibships was 10.7 per cent. 



Obviously the expected standard deviation of simple sampling for 

 comparison must represent samples not of 357 plants each but of 17.85 

 plants each. Now a percentage is obviously a mean (of values all 

 either or 1) . Since "Student" (1908) has shown that the theoretical 

 standard deviation of the mean in samples is given more exactly by 



O" variate ,i -i fvariate 



than by o-mean^ 



(the value for the normal curve conventionally used for the probable 

 error of the mean) and since Ti, the mean size of sample, is small 

 enough to make the correction a matter of considerable importance. 

 V^ — 3 is here used. Since o- variate = VPQ'» we have o-n,ean = 



V 



^^ , where »= 17.85. This gives a theoretical standard devia- 

 n — 3 



tion of 13.0 per cent." 



It is true (Yule, 1911, p. 260) that the ordinary method of calcu- 

 lation of the actual standard deviation is not satisfactory for means 

 when the samples vary in size. A method has been used, however, which 

 obviates this difficulty, so that comparison with the results given by 



is strictly legitimate. Each squared percentage deviation 



v: 



n~3 

 has been weighted by multiplying it by the number of individual 



plants which it represents, and the summation of squared deviations 



has then been divided, not by 2/, the number of samples, but by 



2/ X 'n, the number of samples multiplied by the mean weight or 



average size of sample (in other words, by N, the total number of 



individuals).-^ 



2- In the calculations for table 39 p has been taken as the percentage "iven 

 in this table, to two decimal places, while with all other numbers employed in 

 calculation, including 17 — 3, three or more decimal places have been used as 

 needed. 



23 Algebraic proof of the correctness of the method has kindly been furnished 

 by Frank L. Griffin, Professor of Mathematics, Eeed College. Portland, Oregon. 

 If it develops that this rather obvious device has not been suggested for the 

 purpose, it is to be presented elsewhere with the mathematical proof. When the 

 variates are not grouped in classes the calculation is substantially as easy as 

 without weighting, while the theoretical value is found with much less work 

 than by the method given by Yule (1911, p. 260), which requires the harmonic 

 mean of the sample sizes. 



