THE COLLECTIONS FROM TAHITI AND THEIR INVESTIGATION. 37 
classifying the individual measures of length, those from 16.0 to 16.2 mm. inclusive 
would be combined in a class represented by 16.15 mm.; in classifying the measures 
of width, those from 9.00 to 9.19 would be grouped about the 
class value of 9.10. The individual proportions for 6 diverse | Length. | wiath.| Proportion. 
shells would be as shown herewith: The proportional length 
and width of class values is 56.3 per cent. Now, all of the 16.0 | 9.0 ee 
fractions of a percentage are referred to the half value of | 127 | 691 ise 
that figure, viz, 56.5 per cent, which would represent four of eespleers es 
the six individual measurements given above, but not twoof | 16.2 | 9.1 56.1 
them. On the principle, however, that errors in opposite 
directions balance or neutralize one another in a large series of determinations, the 
discrepancies may be ignored. In the present study, to repeat, the individual propor- 
tions have been seriated and described by their independent constants, primarily to 
obtain the indices of variability. Should anyone demur to this procedure, he may 
obtain the alternative values from the figures relating to characters that were 
directly measured. 
In order to judge whether two component groups of one and the same colony 
are statistically alike, or whether two colonies of the same species which exist in 
neighboring valleys are identical, it is necessary to determine the probable errors 
of the descriptive constants. he probable error of the average (Lm) is 0.67449 Xo, 
divided by the square root of the number of cases. The probable error of the stand- 
ard deviation [Eo] is 0.67449 Xo, divided by the square root of twice the number of 
cases. Observing a numerical difference between the average values of a character, 
in two comparable groups, the difference must be judged as to its significance by its 
own error, which is the square root of the squared probable errors of the two average 
values. Should the difference in question exceed its error, but fall below twice the 
error, it is not significant; should it lie between twice and thrice its probable error, 
then presumably it indicates a real constitutional diversity; when it is greater than 
three times its probable error, then the existence of a real difference is a statistical 
certainty (995 out of 1,000 chances). In the numerical tables, differences that are 
only probable or possible are placed in brackets. It remains to be stated that in 
computing the standard deviation fractions were carried out to eight decimal places, 
in order that the four-place index should be as accurate as possible. 
The length of time required for such quantitative analysis can be estimated 
only by those who themselves have engaged in such work. Even after the lapse of 
weeks and months devoted to classifying and measuring the shells, far more time 
must be occupied in the reduction of the data to the two simple constants of each 
series. [hese figures, together with a single line of text, may be all that represents 
two to eight weeks of mathematical drudgery, however this might be facilitated and 
lightened by the use of numerical tables and the calculating machine. Yet the 
employment of such methods is justified in the final results, which collectively give 
an accurate description of the species and varieties of Partula for the period of 
field-work. 
