SECT. 10.] Theory of the Average. 475 



vantage gained by so doing? The advanced student would 

 of course prefer to work out the answers to these questions 

 by appealing at once to the Law of Error in its ultimate or 

 exponential form. But I feel convinced that the best 

 method for those who wish to gain a clear conception of the 

 logical nature of the process involved, is to begin by treating 

 it as a question of combinations such as we are familiar with 

 in elementary algebra; in other words to take a finite number 

 of errors and to see what comes of averaging these. We can 

 then proceed to work out arithmetically the results of com 

 bining two or more of the errors together so as to get a new 

 series, not contenting ourselves with the general character 

 merely of the new law of error, but actually calculating what 

 it is in the given case. For the sake of simplicity we will 

 not take a series with a very large number of terms in it, but 

 it will be well to have enough of them to secure that our law 

 of error shall roughly approximate in its form to the standard 

 or exponential law. 



For this purpose the law of error or divergence given by 

 supposing our effort to be affected by ten causes, each of 

 which produces an equal error, but which error is equally 

 likely to be positive and negative (or, as it might perhaps 

 be expressed, ten equal and indifferently additive and 

 sub tractive causes ) will suffice. This is the lowest number 

 formed according to the Binomial law, which will furnish to 

 the eye a fair indication of the limiting or Exponential law 1 . 

 The whole number of possible cases here is 2 10 or 1024; that 

 is, this is the number required to exhibit not only all the 

 cases which can occur (for there are but eleven really dis 

 tinct cases), but also the relative frequency with which each 

 of these cases occurs in the long run. Of this total, 252 will 

 1 See, for the explanation of this, and of the graphical method of illus 

 trating it, the note on p. 29. 



