1899] METHOD OF TREATING VARIATIONS 413 



error 1. The proof of this proposition here given is slightly altered 

 from that given in the original, but seems a little clearer. 



Let k be the number of characters, 

 n the total number of variations, 

 I d any deviation of any character, 



v the corresponding variation, 

 ' iv the corresponding probable error, 

 and N the number of variations of any character. 



T , Sum of all the variations _ 1 / Variations of any character \ 



Number of variations k \Number of variations of this character/ 



V _ * ^ ( V \ + V 2 + V 3 • • 



lX - V = k*\ N 



j ,d 1 + d. 2 + d 3 



10 



-I* 



N 



d l + d i) + d. i . . 



" iV r ' 



IV 



1 ^ /Average deviationN 

 k \ Probable error / 



but on the assumption that has been made, if the number of indi- 

 viduals and characters observed be increased, each fraction on the right- 

 hand side of this equation tends to an equality, and from the probability 

 integral — 



Average deviation x _ 1 _ 1 .1 q 

 Probable error '8453 



Hence, the sum of all the variations l divided by the number of varia- 

 tions is equal to 1*18, and this is the standard deviation of a curve 

 whose centre is at and whose probable error is 1. 



Two examples may be given in order to illustrate this conclusion. 

 The first is of a group of 50 herring from the White Sea, and is taken 

 from the work of Prof. Heincke. 



v v 



— for 5 characters, differs from PI 8 by 0"40 

 n 



for these + 3 more, „ „ by - 29 



for 13 other characters ,, ,, by 0*25 



for all (21) „ „ „ by 0-21. 



The difference between theory and observation is evidently very small. 

 In the second example the variations of 54 plaice from St. Andrews 

 Bay are tabulated in full according to the method. Under the second 



1 The terminology used here is that of Galton, Weldon and Pearson. "Deviations" 

 are the observed fluctuations about the average of any character, "Variations" these 

 deviations when expressed in terms of the probable error. These correspond with the terms 

 used by Heincke. 



28 NAT. SC. VOL. XV. NO. 94. 



