SECT. 16.] Averages. 453 



thirty. We then say that the average of this batch of 

 ten is three. Take another set of ten throws, and we may 

 get another average, say four. There is clearly nothing 

 objective peculiarly corresponding in any way to these 

 averages. No doubt if we go on long enough we shall 

 find that the averages tend to centre about 3 5 : we then 

 call this the average, or the probable number of points ; 

 and this ultimate average might have been pretty con 

 stantly asserted beforehand from our knowledge of the con 

 stitution of a die. It has however no other truth or reality 

 about it of the nature of a type : it is simply the limit 

 towards which the averages tend. 



The next class is that occupied by the members of most 

 natural groups of objects, especially as regards the charac 

 teristics of natural species. Somewhat similar remarks may 

 be repeated here. There is very frequently a c limit towards 

 which the averages of increasing numbers of individuals tend 

 to approach ; and there is certainly some temptation to re 

 gard this limit as being a sort of type which all had been 

 intended to resemble as closely as possible. But when we 

 looked closer, we found that this view could scarcely be 

 justified ; all which could be safely asserted was that this 

 type represented, for the time being, the most numerous 

 specimens, or those which under existing conditions could 

 most easily be produced. 



The remaining class stands on a somewhat different 

 ground. When we make a succession of more or less suc 

 cessful attempts of any kind, we get a corresponding series 

 of deviations from the mark at which we aimed. These we 

 may treat arithmetically, and obtain their averages, just as 

 in the former cases. These averages are fictions, that is to 

 say, they are artificial deductions of our own which need 

 not necessarily have anything objective corresponding to 



