368 READINGS IN EVOLUTION, GENETICS, AND EUGENICS 
By the use of this formula we have calculated the standard devia- 
tion (c) of the individuals represented in Figure 62 to be 14.89* 
0.31 scutes. This means that the average deviation from the mean 
is about 14.89 scutes. 
The 0.31 scutes is called the “probable error” and means that 
the figure 14.89 is inaccurate to the extent of being 0.31 scutes too 
high or too low. The probable error is an essential feature of such 
computations, as, without it, we would not be able to rely on the signifi- 
cance of small differences. Suppose, for example, we should find that 
the armadillos of Brazil had a standard deviation of 15.43 0.44 scutes, 
we might conclude that the variability of the Brazilian individuals was 
0.34 scutes greater than that of the Texas individuals. In view of the 
fact, however, that the probable error in one case is+0.31 scutes and 
in the othero.44 scutes we would have to conclude that there was no 
significant difference. In actual practice it has been decided that 
unless the actual difference between two constants is about 4.6 times 
as great as the probable error, the difference is not significant. 
The method of determining the probable error of any calculated 
constant is difficult to understand, but easy to put into practice. For 
example, the formula for calculating the probable error of the standard 
deviation is as follows: 
+0.67450 
a= Von” 
where £ is the probable error, and ” the number of individuals. It 
will be seen that the probability of error diminishes steadily with the 
increase in number of individuals studied. With very large numbers 
the error due to what is known as “‘random sampling’ practically 
disappears. 
BIMODAL AND MULTIMODAL CURVES 
If we confine our biometrical studies to homogeneous populations, 
we get only fairly simple monomodal curves that resemble the normal 
curve of variation, which is a curve of chance; but when we study 
ordinary wild populations, we frequently find that we are dealing with 
a complex of several races, each of which has its own mode and stand- 
ard deviation. Bateson has given us a classic example of this type 
of phenomenon. In studying the length of pinchers in the common 
earwig (Forficulata auricularia), he found that he got a two-humped 
or bimodal curve as shown in Figure 63. It then became evident that 
there were two distinct varieties as figured above. Such studies have 
frequently revealed the heterogeneity of supposedly homogeneous 
