440 Averages. [CHAP. xvm. 



land, or of Englishmen generally with those of Frenchmen, 

 or to ascertain whether the stature of some particular class 

 or district is increasing or diminishing, it really does not 

 seem to matter what sort of average we select provided, of 

 course, that we adhere to the same throughout our investi 

 gations. A very large amount of the work performed by 

 averages is of this merely comparative or non-quantitative 

 description ; or, at any rate, nothing more than this is really 

 required. This being so, we should naturally resort to the 

 arithmetical average ; partly because, having been long in 

 the field, it is universally understood and appealed to, and 

 partly because it happens to be remarkably simple and easy 

 to calculate. 



4. The arithmetical mean is for most ordinary pur 

 poses the simplest and best. Indeed, when we are dealing 

 with a small number of somewhat artificially selected magni 

 tudes, it is the only mean which any one would think of 

 employing. We should not, for instance, apply any other 

 method to the results of a few dozen measurements of lengths 

 or estimates of prices. 



When, however, we come to consider the results of a very 

 large number of measurements of the kind which can be 

 grouped together into some sort of probability curve we 

 begin to find that there is more than one alternative before 

 us. Begin by recurring to the familiar curve represented 

 on p. 29 ; or, better still, to the initial form of it represented 

 in the next chapter (p. 476). We see that there are three 

 different ways in which we may describe the vertex of the 

 curve. We may call it the position of the maximum ordi- 

 nate ; or that of the centre of the curve ; or (as will be seen 

 hereafter) the point to which the arithmetical average of 

 all the different values of the variable magnitude directs us. 

 These three are all distinct ways of describing a position ; 



