Discordant Observations. 369 



we have exaggerated the a priori probability of a mistake. But 

 it may be worth while paying this price for the sake of getting 

 rid of serious mistakes. Especially is this position tenable 

 according to the definition of the qucesitum in the Theory 

 of Errors*, which Laplace countenances. According to this 

 view, the desideratum in a method of reduction is not so 

 much that it should be most frequently right, as that it should 

 be most advantageous ; account being taken, not only of the 

 frequency, but also of the seriousness, of the errors which it 

 incurs. Prof. Stone's method might diminish our chance of 

 being right (in the sense of beiDg within a certain very small 

 distance from the true mark|) ; and yet it might be better than 

 Method I., if it considerably reduced the frequency of large 

 and detrimental mistakes. 



II. (1) (7) Prof, Stone's method is less applicable to the 

 third hypothesis. Though even in this case, if the smaller 

 weights are a priori comparatively rare, it may be safe enough 

 to regard (m — 1) of the m observations as of one and the 

 same type ; and to reject the mth if violently discordant with 

 that supposed type. 



The only misgiving which I should venture to express 

 about this method relates, not to its essence and philosophy, 

 but to a technical detail. Prof. Stone says: — " If we find that 



2 r°° 9 1 



value which makes -7= 1 e~y 2 dy = - [where p is the devia* 

 v 7rjp_ n 



a 



tion of a discordant observation, and a is the modulus of the 

 probability-curve under which the other observations range^ 



and - is the a priori probability of a mistake], all larger 

 values of p are with greater probability to be attributed to 

 mistakes.-" But ought we not rather to equate to -, not the 

 left-hand member of the equation just written, which may be 

 called @(-\ hut 6 m (-\ where m isthe number of observa- 

 tions. ^ I am aware that the point is delicate, and that high 

 authority could be cited on the other side. There is some- 

 thing paradoxical in Cournot's % proposition that a certain 



* See my paper on the " Method of Least Squares/' Phil. Mag. 1883, 

 vol. xvi. p. 3(33 ; also that on " Observations and Statistics," Camb. Phil. 

 Tr. 1885 ; and a little work called ' Metretike ' (London : Temple Co., 1887). 



f The sense defined by Mr. Glaisher, < Memoirs of the Astronomical 

 Society/ vol. xl. p. 101. 



t Exposition de la theorie des Chances, Arts. 102, 114. " Nous ne nous 

 dissimulons pas ce qu'il y a de delicat dans toute cette discussion," I 

 may say with Cournot. 



Phil Mag. S. 5. Vol. 23. No. 143. April 1887, 2 C 



