SECT. 20.] Averages. 457 



them. If the proportions in which the 1000 digits were 

 distributed were the same as those of the 10 the averages 

 would be the same. It is obvious therefore that the arith 

 metical process of obtaining an average goes a very little 

 way towards securing the striking kind of uniformity which 

 we find to be actually presented. 



20. There is another way in which the same thing 

 may be put. It is sometimes said that whatever may have 

 been the arrangement of the original elements the process of 

 continual averaging will necessarily produce the peculiar 

 binomial or exponential law of arrangement. This state 

 ment is perfectly true (with certain safeguards) but it is not 

 in any way opposed to what has been said above. Let us 

 take for consideration the example above referred to. The 

 arrangement of the individual digits in the long run is the 

 simplest possible. It would be represented, in a diagram, 

 not by a curve but by a finite straight line, for each digit 

 occurs about as often as any other, and this exhausts all the 

 arrangement that can be detected. Now, when we con 

 sider the results of taking averages of ten such digits, we see 

 at once that there is an opening for a more extensive arrange 

 ment. The totals may range from up to 100, and there 

 fore the average will have 100 values from to 9 ; and what 

 we find is that the frequency of these numbers is determined 

 according to the Binomial 1 or Exponential Law. The most 

 frequent result is the true mean, viz. 4 5, and from this they 

 diminish in each direction towards and 10, which will each 

 occur but once (on the average) in 10 10 occasions. 



The explanation here is of the same kind as in the former 

 case. The resultant arrangement, so far as the averages are 



1 More strictly multinomial : the efficients of the powers of x in the 

 relative frequency of the different development of 

 numbers being indicated by the co- (l + x + x*+ ... + x 9 ) 10 . 



