368 READINGS IN EVOLUTION, GENETICS, AND EUGENICS 



By the use of this formula we have calculated the standard devia- 

 tion (cr) 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 

 0.54 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 other= fc o.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: 



0.6745* 



Ea- / 



V2H 



where E is the probable error, and n 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 



