SECT. 29.] Theory of the Average. 493 



The black line represents a Law of Error easily stated in 

 words, arid one which, as we shall subsequently see, can be 



conceived as occurring in practice. It represents a state of 

 things under which up to a certain distance from 0, on each 

 side, viz. to A and B, the probability of an error diminishes 

 uniformly with the distance from 0; whilst beyond these 

 points, up to E and F, the probability of error remains con 

 stant. The dotted line represents the resultant Law of Error 

 obtained by taking the average of the former two and two 

 together. Now is the latter better than the former? 

 Under it, certainly, great errors are less frequent and inter 

 mediate ones more frequent ; but then on the other hand 

 the small errors are less frequent : is this state of things on 

 the whole an improvement or not ? This requires us to re 

 consider the whole question. 



29. In all the cases discussed in the previous sections 

 the superiority of the curve of averages over that of the 

 single results showed itself at every point. The big errors 

 were scarcer and the small errors were commoner; it was 

 only just at one intermediate point that the two were on 

 terms of equality, and this point was not supposed to posse* 

 any particular significance or importance. Accordingly we 

 had no occasion to analyse the various cases included under 

 the general relation. It was enough to say that one was 

 better than the other, and it was sufficient for all purposes to 



