36 VARIATION, DISTRIBUTION, AND EVOLUTION OF THE GENUS PARTULA. 
responsible for every direct measurement and for every detail of classification; 
hence the personal coefficient is uniform throughout the entire research. 
The quantitative characters chosen for determination and study are the stand- 
ards of comparative malacology (text-fig. 4). They are (1) the greatest length of the 
shell including its lip (aa to BB); (2) the greatest width, 
also including the lip (cc to pp); (3) the proportions of 
the whole shell, viz, character 2:character 1; (4) length of A 
the outside measurement of the aperture (point F to the 
line BB); (5) width of the aperture (point E to the line pp); 
(6) proportions of the aperture (character 5:character 4); 
(7) the proportion of aperture length to the length of the 
whole shell. Whenever a columellar tooth was present, it 
was noted as a “trace,” as “average,” as “large,” or “very 
large,’ thus giving five classes, including the group in which 8 
the structure was lacking. In the end each adult shell bears 
a distinguishing number corresponding to a place in the Tyyprre. 4.—Diagram of a 
primary tables where its description is given by specific {avila shel to Mlustrate the 
data numbering from Io to 14 or more, as the case may be. 
The dimensions and proportionate measures are then treated by statistical 
methods so as to derive the simplest terms for a collective description of a group; 
it is unnecessary to describe in detail the methods for the reduction of quantitative 
data, in view of their general currency in experimental work. The average or mean 
value is the first constant [M], and this is of course the sum of the classified meas- 
urements divided by the total number; it indicates the typical condition. The 
second constant is the standard deviation, or index of variability [c], which describes 
the degree of conservatism of the members of a group in relation to the average or 
mean condition; the individual deviations from the mean are squared and added, 
and the square root of this sum is the standard deviation. It is expressed in the 
same units as the average (millimeters or per cent). [he measure of absolute vari- 
ability, or coefficient of variation, is often employed in statistical description, as the 
index of variability divided by the mean value and multiplied by roo; in the present 
study, however, this coefficient has not been employed, because of the additional 
labor involved and because it may be readily obtained from the two fundamental 
data as given in the tables, by anyone who wishes to use it. 
It must be added that the individual proportionate measures (characters 3, 
6, and 7) were treated like the direct dimensions, as to many independent variates or 
data. While it is true that the average value of one of them, ¢. g., the proportion of 
the shell, could be obtained by the short method of dividing the group average for 
width by the group average for length, it would be impossible to obtain the standard 
deviation of the relative measurement from other data than the individual deter- 
minations themselves. It happens at times that the average of the proportion does 
not agree exactly with the figure derived from the averages of the lengths and 
widths, for reasons which may best be understood by citing concrete cases. In 
Cc D 
