THE STATISTICAL STUDY OF VARIATION 557 



113 classes and a very irregular and meaningless distribution. On the 

 ordinate we find by tens the numbers of individuals in each class. It 

 will be noted -that the solid line is one connecting the points of inter- 

 section between the class of scute numbers and the number of indi- 

 viduals in these classes. The dotted line represents an ideal fluctuat- 

 ing variation curve, which is practically a mathematical curve of 

 chance. The closeness of fit between the actual and the theoretical 

 curve is very good. The mode is the class including individuals with 

 a scute count of 557-64, and there is a fairly even balance of individuals 

 in the plus and the minus directions. It seems fairly evident from 

 examination of the curve that the individuals with 613 scutes and 

 over are beyond the limits of the theoretical distribution. A further 

 study of these exceptional individuals shows that they are mutations, 

 in which a splitting up of single scutes into paired and twinned scutes 

 has taken place to such an extent as greatly to increase the total num- 

 ber of scutes. 



From the data used in constructing this variation polygon several 

 significant constants may be obtained. The "arithmetical mean" 

 (average number of scutes in the entire 508 individuals) is 558.2. 

 The "median" or halfway point between the extremes is 558. The 

 "mode" or most frequently occurring single type is 557 (the theoreti- 

 cal value being 557.6). 



If we wished to compare a large group of parents with a large group 

 of offspring, or if it were necessary to compare the armadillos of Texas 

 with those of Mexico or Brazil, we could compare them as to mean, 

 median, and mode, and also as to the shape of the polygon of variation. 

 This would give us a very good idea as to whether or not the old species 

 present in these three regions is tending to evolve in different directions 

 under different conditions of life. 



Instead of having to depend on the visual comparison between the 

 variation polygon of two or more different populations, we can reduce 

 the facts about the distribution of the different types about the mean 

 or mode to a simple arithmetical constant, called the "standard 

 deviation," which is usually given the symbol a. This constant is 

 computed as follows: 



■J 3 ? 



/) 



In this formula x represents the deviation of each class from the 

 arithmetical mean; /, the number of individuals in each separate class; 

 2. the sum of all the classes, and «, the total number of individuals 



