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, eg. of the species, have been described by un- 
eritical 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 eed 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, eg. 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 atic of shape and of colour,’ 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). 
