May 20, 1898.] 



SCmNGE. 



687 



very variable. Thus the variability of any 

 curve will be roughly defined by the range, 

 or when the curve is symmetrical, as is 

 usually the case, by the half-range.* 



When the relative frequency of the dif- 

 ferent magnitude-classes gives us the nor- 

 mal curve we may be sure that we are 

 dealing with a single homogeneous group 

 of individuals — a species showing no ten- 

 dency to break up into varieties, or a pure 

 race. But individual measurements do not 



always fall into the normal curve. "We may 

 get any one of a variety of curves such as 

 are shown in Figures 2, 3, 4, 5 and 6. 

 Figure 2 is an asymmetrical curve . Figures 



Figs. 2, 3. 



* The half-range is suggested as the measure of va- 

 riahility on account of the fact that its determination 

 requires no calculation. In all cases in which the 

 curve does not end normally, bat, on the contrary, 

 includes a few highly abnormal individuals, or in 

 cases of groups lying near the line between species 

 and varieties, it would be best to measure divergence 

 in terms of the ' standard deviation.' This quantity 

 is obtained by first finding the mean of all the meas- 

 urements, next getting the deviation of each class 

 from the mean, squaring it, and multiplying it by 

 the number of individuals in the class. Add these 

 products, divide by the number of individuals meas- 



Fir. 6. 



Figs. 4, 5, 6. 



ured, and take the square root of the quotient. These 

 operations are briefly indicated in the formula : 



standard deviation 



n ' 



where M' is the sum of 



the squares of the deviations from the mean and n is 

 the number of individuals. When only half of the 

 curve can be used, find the S d'^ and n for that half. 

 The standard deviation is normally about one-third 

 of the half range. 



