Prof. Edgeworth's Problems in Probabilities. 183 



from below towards the Honour-line and above it. The effect 

 of this consideration is that we have somewhat underrated the 

 probability of displacement for positions above the Honour- 

 line, and somewhat overrated it for positions below. It may 

 be assumed, I think, that these errors will compensate each 

 other when we determine the total number of the uncertain 

 in the manner which I have indicated. 



III. I have made a similar assumption in solving the follow- 

 ing third problem. At any examination of which the circum- 

 stances are given, what number of candidates is most probably 

 displaced ? I proceed as follows. Having ascertained the 

 coefficient of the probability-curve which governs the case, I 

 determine numbers corresponding to equal increments of that 

 coefficient above and below the Honour-point. In each of 

 the degrees so constituted I find how many candidates are 

 comprehended ; and I assign to the candidates in each degree 

 the probability of displacement which is found by Problem I. 

 (with the aid of the proper tables) to appertain to the centre 

 of the degree. The product of the number of candidates and 

 the probability of their displacement gives for each degree 

 the number most probably displaced ; which numbers being 

 added give the total number most probably displaced. It is 

 assumed that the underrating of the probability above the line 

 will fairly well be compensated by the overrating below. 



An example will make my meaning clear. With reference 

 to an examination for 50 clerkships of the second class, of 

 which the statistics are given in the Twentieth Report of the 

 Civil Service Commissioners*, how many would most pro- 

 bably be displaced if the work has been marked by another 

 set of equally competent examiners. The problem is of the 

 species Bayz, the candidates being numerous and the papers 

 about ten in number. The Honour-line is at 1720, and the 

 probable error for the regulating Probability- curve (what I 

 have elsewhere called the probable discrepancy) is taken as 

 50 ; upon the assumption that the Mean Error for each of 

 the ten papers is 15 per cent., that is the lowest coefficient 

 which I have actually observed. Accordingly the intervals 

 1720-1730 &c. correspond each to a fifth of the Probable 

 error "j*. The computation is shown in the annexed Table. 



* Parliamentary Papers, 1876, xxii. p. 180. 



t Here called probable error with reference to the tables in the books . 

 elsewhere in connexion with the subject-matter probable discrepancy .' 

 being \/2x probable divergence of a mark from the "true mark." 



