and the Elimination of Chance. 309 



observation. It is thus that Laplace determines the proba- 

 bility that the difference of a millimetre between the means 

 of two sets of barometrical observations made at 9 a.m. and 

 4 p.m. respectively is due to a " constant cause," rather than 

 mere chance. I have applied the method to a great variety 

 of statistical inferences in a paper read before the Statistical 

 Society this year. 



In that paper *, and elsewhere, I have discussed the general 

 principles upon which the method is based. The subject of 

 this paper is the particular case where the observed divergence 

 is very great — so great as to raise a doubt whether the law of 

 error upon which the whole theory turns is adequately ful- 

 filled. For instance, in the example just cited from Laplace, 

 the observed divergence much exceeds (as will presently f 

 appear) the limit analogous to that which is ordinarily postu- 

 lated in the application of the law of error to Bernouilli's 

 theorem. It is attempted here to remove this scruple by 

 proving the following propositions. The received formula is in 

 general sufficiently accurate, or at least safe, in that it affords 

 a superior limit to the probability of mere accident, an a for- 

 tiori argument in favour of law. But in a certain class of 

 cases correction is required, and is attainable in a certain 

 species of that class. 



The subject may be divided according to the presence or 

 absence of certain properties which frequently occur, and 

 whose occurrence tends to the correctness or at least the 

 corrigibility of the received formula. These properties are 

 (a) the symmetry, (/3) the finiteness, and (7) the binomial 

 character of that facility-curve, or more generally -locus, 

 which represents the possible errors of an individual obser- 

 vation, or more generally of each of the elements^ of which the 

 sum is our datum. The negatives of these attributes may be 

 denoted thus : — a, {3, y . 



a ft y. According to this arrangement, the first case to 

 be considered is that in which all the properties are present. 

 This is that simple case of Bernouilli's theorem where the 

 probability of both alternatives is the same, namely J. The 

 interest of this theorem greatly transcends games of chance. 

 For, as Mr. Galton has pointed out in this Journal §, the laws 



* " Methods of Statistics," Journal of Statistical Society, Jubilee Num- 

 ber, 1885. " Observations and Statistics," Cambridge Phil. Soc. 1885. 



" On the Method of ascertaining Variations " read before the 



British Association, 1885 ; published in the Journal of the Statistical 

 Society, January 1886 ; Proceedings of the Society for Psychical Research, 

 parts viii. and x. 



t Below, pp. 311, 317. % See note to p. 308. 



§ Phil. Mag. Jan. 1875. 



