458 Averages. [CHAP, xviii. 



concerned, is only necessary in the sense that it is a neces 

 sary result of certain physical assumptions or experiences. 

 If all the digits tend to occur with equal frequency, and if 

 they are independent (i. e. if each is associated indifferently 

 with every other), then it is an arithmetical consequence 

 that the averages when arranged in respect of their magni 

 tude and prevalence will display the Law of Facility above 

 indicated. Experience, so far as it can be appealed to, shows 

 that the true randomness of the selection of the digits, i.e. 

 their equally frequent recurrence, and the impartiality of 

 their combination, is very fairly secured in practice. Ac 

 cordingly the theoretic deduction that whatever may have 

 been the original Law of Facility of the individual results 

 we shall always find the familiar Exponential Law asserting 

 itself as the law of the averages, is fairly justified by ex 

 perience in such a case. 



The further discussion of certain corrections and refine 

 ments is reserved to the following chapter. 



21. In regard to the three kinds of average employed 

 to test the amount of dispersion, i.e. the mean error, the 

 probable error, and the error of mean square, two im 

 portant considerations must be borne in mind. They will 

 both recur for fuller discussion and justification in the course 

 of the next chapter, when we come to touch upon the Method 

 of Least Squares, but their significance for logical purposes 

 is so great that they ought not to be entirely passed by at 

 present. 



(1) In the first place, then, it must be remarked that in 

 order to know what in any case is the real value of an error 

 we ought in strictness to know what is the position of the 

 limit or ultimate average, for the amount of an error is 

 always theoretically measured from this point. But this is 

 information which we do not always possess. Recurring 



