374 



Mr. D. McAlister on 



[Nov. 20, 



9. We may next show that if the Arithmetic Mean (A.M.) of all the 

 measures be formed then 



4— (¥> 



This follows strictly when the nnmber of measures is indefinitely 

 great. It still gives a good approximation to the value of In when the 

 number is considerable. 



10, Among the measures greater than the mean there is one which 

 may be called middlemost : i.e., such that it is an even wager that a 

 measure (greater than the mean) lies above it or below it. A similar 

 middlemost measure exists among those that are less than the mean. 

 As these two measures, with the mean, divide the curve of facility 

 into four equal parts, I propose to call them the " higher quartile " 

 and the " lower quartile " respectively. It will be seen that they 

 correspond to the ill-named "probable errors " of the ordinary theory. 

 If Q, q be the quartiles we can show that 



h log -§-=•4769 . . . =-Mog?, 

 a a 



so that Qq=a 2 . Thus also 



0*06208 . . )=a h =q h (l'6ll . . ). 



Similarly between zero and lower quartile we place a mid-measure 

 which we call the " lower ocbile." The " higher octile " will subdivide 

 the interval between the higher quartile and infinity. If 0, o, be the 

 octiles, we have — 



0*(-443)=a*=o* (2-255 . . ): 

 :as before Oo=a 2 : and 



:1'40 . . . 



Q 



It will be observed that in the curve of distribution Q and q are 

 ordinates equidistant from the mean and the terminal ordinates. is 

 equidistant from Q and the asymptote. 



The analogues suggested by the mean error, and mean square 

 error, &c, of the ordinary theory have no very practical value or 

 significance for us. It will be remembered that they are introduced 

 to obviate the difficulties arising from negative errors. In our 

 problem these somewhat artificial functions have no special place.* 



* The mean-square-measure is a exp. (^jp)- ^he geometric mean of the measures 

 greater than mean is 3 exp. ( 564 m _ j_\ 



