12 THE MEASUREMENT OF VARIATION. 



acted on 190 times by 18 favourable and 2 unfavour- 

 able, 15,504 times by 15 favourable and 5 unfavour- 

 able, and no less than 184,756 times by 10 favourable 

 and 10 unfavourable. Let us consider that the organ- 

 isms acted on by 20 favourable and unfavourable 

 agencies have their size increased by 20 per cent., those 

 acted on by 15 favourable and 5 unfavourable by 15 

 5 = 10 per cent., and so on. If now these percentage 

 increments and decrements be plotted out at equal dis- 

 tances on a base line, and ordinates corresponding to 

 the theoretical frequencies erected from each, then by 

 joining these ordinates we shall obtain a curve which is 

 practically identical in form with the dotted line curve 

 given in Fig. 3; i. e., with the probability curve of the 

 law of frequency of error. Thus, by a simple arith- 

 metical method, we can obtain a series approximating 

 more and more closely to the probability curve, the 

 greater the number of times the expression (J + J) is 

 expanded. Expanded 20 times, the average error is 

 less than .5 per cent., and for a greater number of times 

 it becomes rapidly smaller and smaller. 



The deviations in the dimensions of organisms are 

 thus distributed about their mean in a symmetrical 

 manner, in accordance with the law of frequency of 

 error. This is true not of one or two characteristics 

 of an organism, but probably, in the majority of cases, 

 of nearly all of them. The dependence of variation on 

 the Laws of Probability was first demonstrated by Que- 

 telet * in the case of height and chest measurements of 

 soldiers. These he showed to group themselves in ac- 

 cordance with the ordinates of a binomial curve. 

 * " Lettres sur la theorie des probabilites," Brussels, 1846. 



