SECT. 21.] Averages. 459 



once more to the three principal classes of events with which 

 we are concerned, we can readily see that in the case of 

 games of chance we mostly do possess this knowledge. In 

 stead of appealing to experience to ascertain the limit, we 

 practically deduce it by simple mechanical or arithmetical 

 considerations, and then the error in any individual case or 

 group of cases is obviously found by comparing the results 

 thus obtained with that which theory informs us would ulti 

 mately be obtained in the long run. In the case of de 

 liberate efforts at an aim (the third class) we may or may 

 not know accurately the value or position of this aim. In 

 astronomical observations we do not know it, and the method 

 of Least Squares is a method for helping us to ascertain it as 

 well as we can ; in such experimental results as firing at a 

 mark we do know it, and may thus test the nature and 

 amount of our failure by direct experience. In the remain 

 ing case, namely that of what we have termed natural kinds 

 or groups of things, not only do we not know the ultimate 

 limit, but its existence is always at least doubtful, and in 

 many cases may be confidently denied. Where it does exist, 

 that is, where the type seems for all practical purposes per 

 manently fixed, we can only ascertain it by a laborious resort 

 to statistics. Having done this, we may then test by it the 

 results of observations on a small scale. For instance, if we 

 find that the ultimate proportion of male to female births is 

 about 106 to 100, we may then compare the statistics ot 

 some particular district or town and speak of the consequent 

 error, viz. the departure, in that particular and special 

 district, from the general average. 



What we have therefore to do in the vast majority of 

 practical cases is to take the average of a finite number of 

 measurements or observations, of all those, in fact, which 

 we have in hand, and take this as our starting point in 



