10 THE MEASUREMENT OF VARIATION. 



tunus depurator. In order to get rid as far as possible 

 of the factor of size, and obtain a measure of the varia- 

 bility apart from this, each measurement was calculated 

 as a fraction on that of the carapace length of the 

 crab taken as 1000. The numbers on the abscissa line 

 therefore represent 1230, 1240, etc., thousandths of 

 the total length. The figures on the central ordinate 

 represent the numbers of individuals of each particular 

 dimension. For instance, one may gather that 16 indi- 

 viduals had a post-spinous length of 1260, 172 of them 

 one of 12 97, and so on. 



If this curve be compared with the general contour 

 of the previous figure, it will be seen at a glance that 

 there is a much more regular rise and fall, especially in 

 regard to the extreme measurements. In fact, it does 

 not differ very greatly from the dotted line curve upon 

 which it is superposed, and supposing the number of 

 observations had been greater, one would expect the 

 approximation to be still closer; supposing it had been 

 infinitely great, one would expect the two curves to be 

 identical. Now this dotted line is a probability curve, 

 or a diagrammatic representation of the Law of Fre- 

 quency of Error, of which the mathematical expression* 

 was first deduced by Gauss at the beginning of the 

 last century. It would be out of place to attempt 

 to reproduce its mathematical proof here, but perhaps 

 a concrete instance may help to bring home to the non- 

 mathematical reader the fact that variability does obey 



* This expression is y = ke -W a , or taking k and h each as unity, 

 y = s , where e is the base of Naperian logarithms, and y an ordi- 



6 " 



nate erected from any point on the abscissa, distant x from the 

 middle ordinate. 



