SECT. 18.] Theory of the Average. 483 



of distribution will give a result which bears the same 

 general relation to the individual values that the dotted lines 

 above bear to the black line 1 . 



18. This being so, the speculative advantages of one 

 method of combining, or averaging, or reducing, our observa 

 tions, over another method, irrespective, that is, of the 

 practical conveniences in carrying them out, will consist 

 solely in the degree of rapidity with which it tends thus to 

 cluster the result about the centre. We shall have to subject 

 this merit to a somewhat further analysis, but for the present 

 purpose it will suffice to say that if one kind of average gave 

 the higher dotted line in the figure on p. 479 and another 

 gave the lower dotted line, we should say that the former 

 was the better one. The advantage is of the same general 

 kind as that which is furnished in algebraical calculation, by 

 a series which converges rapidly towards the true value as 

 compared with one which converges slowly. We can do 

 the work sooner or later by the aid of either; but we get 

 nearer the truth by the same amount of labour, or get as 

 near by a less amount of labour, on one plan than on the 

 other. 



As we are here considering the case in which the indi 

 vidual observations are supposed to be grouped in accordance 



1 Broadly speaking, we may say ingly rapid, that is, when the ex- 



that the above remarks hold good of treme errors are relatively very few, 



any law of frequency of error in it still holds good. But if we were 



which there are actual limits, how- to take as our law of facility such an 



ever wide, to the possible magnitude TT . . , . 



equation as y = - , (as hinted by 

 of an error. If there are no limits to 



the possible errors, this characteristic De Morgan and noted by Mr Edge- 



of an average to heap its results up worth : Cainb. Phil. Trans, vol. x. 



towards the centre will depend upon p. 184, and vol. xiv. p. 160) it does 



circumstances. When, as in the ex- not hold good. The result of aver- 



ponential curve, the approximation aging is to diminish the tendency to 



to the base, as asymptote, is exceed- cluster towards the centre. 



31 2 



