456 Averages. [CHAP. xvm. 



arithmetic, and one therefore which might be asserted a 

 priori ? 



Whatever force there may be in the above objection arises 

 principally from the limitations of the example selected, in 

 which the number chosen was so large a proportion of the 

 total as to exclude the bare possibility of only extreme cases 

 being contained within it. As much confusion is often felt 

 here between what is necessary and what is matter of ex 

 perience, it will be well to look at an example somewhat 

 more closely, in order to determine exactly what are the 

 really necessary consequences of the averaging process. 



19. Suppose then that we take ten digits at random 

 from a table (say) of logarithms. Unless in the highly un 

 likely case of our having happened upon the same digit ten 

 times running, the average of the ten must be intermediate 

 between the possible extremes. Every conception of an 

 average of any sort not merely involves, but actually means, 

 the taking of something intermediate between the extremes. 

 The average therefore of the ten must lie closer to 4 5 (the 

 average of the extremes) than did some of the single digits. 



Now suppose we take 1000 such digits instead of 10. We 

 can say nothing more about the larger number, with de 

 monstrative certainty, than we could before about the smaller. 

 If they were unequal to begin with (i. e. if they were not all 

 the same) then the average must be intermediate, but more 

 than this cannot be proved arithmetically. By comparison with 

 such purely arithmetical considerations there is what may be 

 called a physical fact underlying our confidence in the grow 

 ing stability of the average of the larger number. It is that 

 the constituent elements from which the average is deduced 

 will themselves betray a growing uniformity : that the pro 

 portions in which the different digits come out will become 

 more and more nearly equal as we take larger numbers of 



