450 Averages. [CHAP. xvni. 



taking their (arithmetical) average, provided only they are 

 as often in excess as in defect 1 . For this purpose all that is 

 necessary is that equal excesses and defects should be 

 equally prevalent. It is not necessary to know what is the 

 law of variation, or even to be assured that it is of one par 

 ticular kind. Provided only that it is in the language of 

 the diagram on p. 29, symmetrical, then the arithmetical 

 average of a suitable and suitably varied number of measure 

 ments will be free from this source of disturbance. And 

 what holds good of this cause of variation will hold good of 

 all others which obey the same general conditions. In fact 

 the equal prevalence of equal and opposite errors seems to 

 be the sole and sufficient justification of the familiar process 

 of taking the average in order to secure accuracy. 



14 We must now make the distinction to which at 

 tention requires so often to be drawn in these subjects 

 between the cases in which there respectively is, and is not, 

 some objective magnitude aimed at : a distinction which the 

 common use of the same word &quot; errors &quot; is so apt to obscure. ! 

 When we talked, in the case of the brass rod, of excesses 

 and defects being equal, we meant exactly what we said, viz. 

 that for every case in which the true length (i. e. that de- 1 

 termined by the authorized standard) is exceeded by a given 

 fraction of an inch, there will be a corresponding case in 

 which there is an equal defect. 



On the other hand, when there is no such fixed objective 

 standard of reference, it would appear that all that we mean; 

 by equal excesses and defects is permanent symmetry of 

 arrangement. In the case of the measuring rod we were 



1 Practically, of course, we should take this variation as a specimen of... 



allow for the expansion or contrac- one of those disturbances which may 



tion. But for purposes of logical be neutralised by resort to an ave-J 



explanation we may conveniently rage. 



