SECT. 11.] Averages. 447 



the probable error on each side, whilst the middle one will 

 mark the median. This median, as was remarked, coincides, 

 in symmetrical curves, with the arithmetical mean. 



It is best to stand by accepted nomenclature, but the 

 reader must understand that such an error is not in any 

 strict sense probable. It is indeed highly improbable that 

 in any particular instance we should happen to get just this 

 error : in fact, if we chose to be precise and to regard it as 

 one exact magnitude out of an infinite number, it would be 

 infinitely unlikely that we should hit upon it. Nor can it be 

 said to be probable that we shall be within this limit of the 

 truth, for, by definition, we are just as likely to exceed as to 

 fall short. As already remarked (see note 011 p. 441), the 

 maximum ordinate would have the best right to be regarded 

 as indicating the really most probable value. 



11. (5) The error of mean square. As previously 

 suggested, the plan which would naturally be adopted by 

 any one who had no concern with the higher mathematics of 

 the subject, would be to take the mean error for the pur 

 pose of the indication in view. But a very different kind of 

 average is generally adopted in practice to serve as a test of 

 the amount of divergence or dispersion. Suppose that we 

 have the magnitudes x v x v ...x n \ their ordinary average is 



1 ( x + x + +x\ and their errors are the differences 

 n ^ 1 



between this and x v x v . . .x n . Call these errors e lt e v . . .e n , then 

 the arithmetical mean of these errors (irrespective of sign) is 



I ( e 4. e + + e } The Error of Mean Square 1 , on the other 



n ^ l 2 



hand, is the square root of - (e? + e? +...+ O- 



i There is some ambiguity in the commonly uses the expression Error 

 phraseology in use here. Thus Airy of Mean Square to represent, as 



