SECT. 23.] Averages. 461 



an average, and that every average necessarily rejects a 

 mass of varied detail by substituting for it a single result. 

 We had, say, a lot of statures : so many of 60 inches, so 

 many of 61, &c. We replace these by an average of 68, 

 and thereby drop a mass of information. A portion of this 

 we then seek to recover by reconsidering the errors or 

 departures of these statures from their average. As before, 

 however, instead of giving the full details we substitute an 

 average of the errors. The only difference is that instead of 

 taking the same kind of average (i.e. the arithmetical) we 

 often prefer to adopt the one called the error of mean 

 square. 



23. A question may be raised here which is of sufficient 

 importance to deserve a short consideration. When we have 

 got a set of measurements before us, why is it generally 

 held to be sufficient simply to assign: (1) the mean value; 

 and (2) the mean departure from this mean ? The answer 

 is, of course, partly given by the fact that we are only sup 

 posed to be in want of a rough approximation : but there is 

 more to be said than this. A further justification is to be 

 found in the fact that we assume that we need only con 

 template the possibility of a single Law of Error, or at any 

 rate that the departures from the familiar Law will be but 

 trifling. In other words, if we recur to the figure on p. 29, 

 we assume that there are only two unknown quantities or 

 disposable constants to be assigned ; viz. first, the position of 

 the centre, and, secondly, the degree of eccentricity, if one 

 may so term it, of the curve. The determination of the 

 mean value directly and at once assigns the former, and the 

 determination of the mean error (in either of the ways re 

 ferred to already) indirectly assigns the latter by confining us 

 to one alone of the possible curves indicated in the figure. 

 Except for the assumption of one such Law of Error the 



