328 GEORG DUNCKER [NOVEMBER 
tion. This done, the results may be further dealt with. Firstly, all 
the observed variants have to be arranged in series according to their 
numerical value, and the frequency of each among the (7) investigated 
individuals has to be determined. Thus we get the empirical series of 
variation of the character in individuals. Weldon [20], for example, 
counted the number of the dorsal rostral teeth in 915 individuals of 
Palaemonetes varians, whence he got 
Variants foe | 2 3 4 5 6 —_.7 (rostral teeth) 
Frequencies . 2 18 123 372 349 50 1 (individuals). 
A series of variation may be graphically represented by noting the 
variants in the order of their numerical values as points of equal 
distances on an abscissa,.and by erecting ordinates from each of these 
points which represent by their length the relative (percentage) 
frequencies of the corresponding variants. Straight lines drawn 
between the free ends of each two adjacent ordinates, together with 
the abscissa, will give the outline of the polygon of variation of the 
character. The average value of the character, that is, the arithmetical 
mean of the variants, corresponds to a point on the abscissa (M); the 
ordinate erected to the latter is the centroid vertical of the polygon 
(y,.). The summit of the polygon of variation usually lies near the 
centroid vertical; its variant has been called the mode; but the 
mode is neither more “typical” nor more “normal” than any other 
variant existing. 
The single ordinates of frequency are generally lower the more 
distant they are from the centroid vertical. The polygon of variation 
is broad and low when there is great variability in the character, 
but high and narrow in the opposite case. The best and simplest 
expression of the degree of variability of a given character is the 
square root of the average square deviations of its variants from 
the mean. This value, which may be called the index of variability 
of the character (e), corresponds to a piece of the abscissa; it is 
expressed by the same unit as the variants of the character. The 
above-cited series of variation of Palaemonetes has the index of 
variability, °8627 rostral teeth. 
While the average values of a character may differ widely in 
different form-units of the same species, the indices of variability 
remain fairly constant * not only in the form-units of the same species, 
1 Examples : 
I. Number of fin rays in dorsal fin anal fin 
M € M € 
of Plewronectes flesus, Baltic 39°46 1°4838 
i ,, North Sea 41°56 1°7739 
4 a ,, Plymouth 61°7214 2°3895 43°6098 1°6026 
a ne americanus (BuMPUS [4]) 65°06 2°4467 48°62 1:8188 
,, Rhombus maximus (PETERSEN [14]) 62°98 2°2533 45°86 1°6792 
II. Number of rostral teeth dorsal ventral 
in Palaemonetes varians (WELDON [20]) 4°3137 0°8627 1°6948 0°4799 
a x vulgaris 8:2819 0°8145 2°9781 0°4477 
