Mr. F. Y. Edgeworth's Problems in Probabilities. 377 

 since 



k=\/j!L_c, and Q'=-477c 



V 1-7 n ' 



approximately. The odd terms above the third vanish, and 

 the even terms may be neglected, involving the higher powers 



of k 2 . i.e. of -. 

 n 



In this reasoning we have supposed the Quartile to be 



determined by taking the point which divides the whole set 



of given observations into two groups whose numbers are to 



each other as three to one. If we determined the Quartile 



by taking the point which divided - observations above the 



Median into two groups each numbering -., the conclusion 



would be much the same except that, instead of n } we must 



n 

 put <j, and put 1*7 for 2 in the value of k. 



As an example of the last case, let us take n=80; that 

 being, I think, the greatest number of independent observa- 

 tions to which the series of returns of Notes in the hands of 

 the public between 1826-1844 can be regarded as equivalent. 

 And let a = 4. Then the correction which is to be made 

 upon the prima facie solution #(4x*477) or *993 is *008, 

 or about *01. 



This conclusion may be roughly verified by the following 

 table, in which the first column represents several hypotheses 

 as to the relation between Q' the length of the (apparent) 

 Quartile measured from the Mean (supposed given) and c 

 the Modulus of the given observations. 



These hypotheses are thus constituted. The central hypo- 

 thesis corresponds to the case in which a single observation is 

 as likely to fall outside as inside Q'. According to the hypo- 

 thesis immediately above the central one (Q' = *60c), the 

 probability of a single observation falling inside Q' is *6. 

 According to the hypothesis next above, the corresponding 

 probability is *7. Conversely, below the centre the corre- 

 sponding probabilities are *4, *3, &c. According to received 

 principles * it is allowable to regard each of these hypotheses 

 as a priori equally probable. 



* See my paper on d, priori Probabilities (Phil. Mag. 1884, Sept.) ; also 

 that on " Observations and Statistics " (Camb. Phil. Trans. 1885). It 

 might have seemed more natural, though not really, I think, preferable, 

 to take as the equi-probable hypotheses the equations of Q' to equicrescent 

 values of c (or of c to equicrescent values of Q,'). I have done so in con- 

 structing analogous tables for the solution of problem iii. (below, p. 381). 

 Phil. Mag. S. 5. Vol. 22. No. 137. Oct. 1886. 2 



