1899] METHOD OF TREATING VARIATIONS 415 



From this theoretical conclusion we are led on to the practical out- 

 come. If the conditions are altered we get a second group of 

 individuals each of which conforms to the same law, but the centre of 

 the averages is altered. Hence if any individual be chosen at random 

 from one of these groups we should be able to tell by an examination 

 of its characters to which group it belonged. Hence the following 

 practical rule. 



3. The sum of the squares of the variations in the characters of 

 a certain group is a minimum for the individuals of that group. 



This follows directly from the equation of the probability integral, 



V 



* sjir 



the nearer y approaches the centre of the curve, or the average, the 

 smaller x 2 becomes. This being true for all the characters, we have 

 that *Zx 2 is a minimum for the variations in the characters of a certain 

 group. Hence, if it is desired to know to which of several known 

 groups a certain individual belongs, it is necessary to calculate the 

 variations of each character of the individual from the respective 

 averages of the several groups, then find the sum of the squares of 

 these variations, and the least sum shows the group to which the 

 individual is most nearly allied. The more characters that are taken 

 the more likely is the result to be right, but less characters are neces- 

 sary the greater the number of individuals. The first example taken 

 to illustrate this is one of several given by Prof. Heincke. It refers to 

 a single specimen of the group of herring obtained from the White 

 Sea, which had 58 vertebrae where the average was 53'6. One might 

 think, therefore, that this individual was abnormal for this group, or 

 belonged to quite another group. Two other groups are therefore 

 taken, the one from the west and south-west coast of Norway (Vaarsild), 

 which has 57*5 as the average number of vertebrae, the other from 

 the Jutland Bank off Denmark, which has 56*6 vertebrae on the 

 average. When other characters are considered, however, and the 

 variations of this single individual form the averages of the three 

 groups calculated according to the method, we find that 



From the average of the White Sea (35 characters) . . (x 2 or)v 2 = 3"213 



,, ,, Vaarsild (35 characters . . . v 2 = 3'696 



,, ,, White Sea (37 characters) . . -t» 2 = 3*225 



,, ,, Jutland Bank (37 characters) . w 2 = 3 - 617 



In each case the least value shows that this individual more closely 

 approaches to the herring of the White Sea in spite of its having a 

 seemingly abnormal number of vertebrae. From this it follows that 

 whilst in one character an individual may be very much above the 

 average, it has a variation or group of variations in other characters 

 below the average, which balance by " defect " what the first has in 

 '" excess." 



