1899] VARIATION- STATISTICS IN ZOOLOGY 327 



not only several variants, but also their relative frequencies. Until 

 now this necessity has generally been neglected. The characters of a 

 group of individuals, e.g. of the species, have been described by un- 

 critical generalisation of single results which were regarded as 

 " typical " or " normal," or by average values got mostly from small 

 numbers of observations, which naturally represent only idealised 

 single results, or, in the best cases, by so-called " ranges of variation." 

 The latter are merely chance results of observation without definitive 

 value ; they show the group to be variable, without indicating the 

 manner of its variation. The only quantitative data we occasionally 

 meet with are indefinite terms, such as " frequent " or " rare." 



For determining not only the variants of comparable objects, but 

 their frequencies as well, we must use statistics. Statistics are collec- 

 tions of single data, brought together according to certain points of 

 view, of qualitative differences of numerous objects belonging to the 

 complex to be investigated, and of the frequencies with which these 

 differences occur. 



In order to investigate the variation of any character of a form- 

 unit, the character in question has to be determined in as many 

 individuals of the form-unit as possible, the variants in which the 

 character is found are to be noted, and finally the frequency with 

 which each of these variants occurs, is to be determined. This 

 method can be applied to every character, to conditions of shape and 

 colour as well as to dimensional or numerical conditions of organs. 



Then the first result as regards variation will be, that if the 

 number of individuals investigated is not too small, the relative 

 frequencies of the single variants of the character will be nearly 

 constant in each lot of the same form-unit. For instance, when a 

 character has been investigated thrice, each time in 500 individuals, 

 and in all cases nearly equal percentages of its variants have been 

 found, we may conclude, according to the law of great numbers, that 

 in the whole form-unit also the variants are distributed in the same 

 proportion. Secondly, closely allied form-units, e.g. the two sexes of 

 the same breed and in a similar stage of development, may possibly 

 agree in the mean and range of a character, and yet sensibly differ in 

 the frequency distribution of its variants. Such differences of com- 

 plexes of individuals are only to be made out by statistically examining 

 the variability of their characters. 



The statistical investigation of such characters as cannot be 

 numerically expressed, like conditions of shape and of colour, 1 cannot go 

 beyond this point. But in numerical characters, such as dimensions 

 or numbers of homologous organs, the variants represent numbers 

 which differ by constant values, the units of dimension or enumera- 



1 At present there is an increasing tendency to express numerically these conditions 

 also ; thus Davenport seeks to numerically determine colour- variations by the ' ' colour- 

 wheel" (Science, N.S. vol. ix. No. 220, p. 415-416). 



