Raymond Pearl 
59 
This table brings out a number of points of interest, but before considering it 
in detail it is necessary to have before us the values of the probable errors of 
certain of the constants, on the assumption that all the distributions obey the 
normal or Gaussian law. The formulae for these probable errors have been given 
by Pearson (1905 and elsewhere), and it is unnecessary to repeat them here. In 
Table IV are given the values of the probable errors of the four constants which 
are of the most importance in testing whether a distribution significantly differs 
from the normal law, viz., V/^i, A, skewness, and the " modal divergence," d. 
TABLE IV. 
Probable Errors of Constants for Normal Distribution. 
Constant 
Series A. N = 201 
Series B. N==175 
^2 
Skewness 
d Length 
d Breadth 
+ ■1165 
+ -2331 
± -0583 
+ ■1532 mikrons 
±•0615 „ 
+ ■1249 
+ -2498 
± ^0624 
+ ■1589 niikrons 
± -0734 
Examining the values given in Table III in connection with those for the 
probable errors in Table IV we see at once a number of differences between 
Series A and Series B. Considering first the question of the symmetry of the 
distributions, it is evident, from the values of a/A ^^^^ of the skewness, that for 
Series A the distributions of both length and breadth are symmetrical within the 
limits of the errors of random sampling. In both distributions the skewness and 
V/3i differ from their theoretical value (if the distribution be truly symmetrical) 
of zero, by only small fractions of their probable errors. With Series B the case 
is different : here both the length and breadth distributions give values for v'A 
and skewness which differ from zero by more than their probable errors. In the 
case of the breadths this deviation rises to more than twice the value of the 
probable error. It is probable that we have to do with real skewness here, and 
not simply with an effect of random sampling. An examination of the " modal 
divergence " leads to the same result : namely, in both the length and breadth 
distribution of Series A the mode does not significantly differ from the mean, 
while in Series B the value of d is for both distributions greater than its probable 
error. For the breadths this divergence of d from zero is about 2'6 times its 
probable error. The skewness is positive in both of the Series B distributions, 
or the mean is greater than the mode. 
Turning to the kurtosis (cf. Pearson, 1905, p. 173) measured by the quantity 
t; = — 3, it is seen that for the lengths in Series A it has a value of — "4343, 
with a probable error (if the distribution were truly mesokurtic) of + "2331. We 
conclude then that the distribution is probably significantly leptokurtic (i.e. is 
less flat-topped than the normal curve), and that we shall get better results if we 
8—2 
