No. 539] DISTRIBUTION OF PURE LINE MEANS 689 



Roemer does not give us the population standard 

 deviations for the several characters in 1909 but only the 

 averages, m lf m 27 m 3 , • • m s , and the standard deviations 

 °i> °"2> °3> ■*•> °>- We may approximate the desired con- 

 stants very closely indeed-"' by the following method. 



Let there be s samples or pure lines of n u n 2 . w 3 , n 8 

 individuals each, with means m 19 m 2y m 3 , m,, and 

 standard deviations <r,, o- 2 , o- 3 , o- s . These form the 

 population S(n) —N, for which the physical constants 

 5 and M are desired. 



The mean is clearly M = S(nm)/S(n). 



In calculating the S.D. we may take the first two 

 rough moments, v/, v 2 \ about any point we please and 

 adjust by the familiar formula o 2 =fi 2 =v 2 / — v/ J . If 

 the moments be taken about 6 v x ' = M, and it is at once 

 clear that for the population 



when 8 indicates a summation for all groups or lines. 7 

 The population constants have been calculated by these 

 formula? for all the characters of Roemer's two large 

 scries. He lias given population constants. M and 5, 

 for the 1908 series, the parents of the 1909 plants. 



The two are conveniently laid side by side for com- 

 parison in Table I. 8 The data in hand hardly seem to 

 justify detailed comparison with reference to probable 



6 There is no approximation in the formula. The accuracy in practise 



upon the number of decimal places retained in the arithmetical routine. 



•For several advantages in doing this see Amkr. Nat., Vol. 44, pp. 693- 

 699. 1910. 



ingly laborious, involving as it does the determination and summation by 

 pairs of 3,108 squares, and the summation of the products of their totals 



test of his results to a few hours. ? ? 7 



8 The constants for 1908 arc taken from Eoemer's Table L Those for 



