Prof. Edo-ewortri's Problems in Probabilities. 187 



76th which is 1597; in round numbers 1600. In the seventh 

 column, each entry is the discrepancy which is as likely as not to 

 occur between the sum of ten marks given to any candidate's 

 papers at the examination referred to and the sum of the marks 

 which might have been given to the same work by any other 

 set of equally competent examiners. This figure is calculated 

 from the formula: Mean Discrepancy = Mean Error x \/2x*477 

 (or Mean Error x "67 ...)-*- s/ 10. Here 10 is the number (or 

 greater than the number) of the papers answered by a candidate. 

 The other figures are explained in the books on Probabilities. 

 The Mean Error is a coefficient determined by observation in the 

 manner described in this paper (above 176). Por most of the 

 examinations the lowest figure actually observed, viz. 15, has been 

 taken for the Mean Error. Por the India Civil Service the 

 higher coefficient 18 has been used ; partly because that figure 

 actually has been observed for that examination ; and partly because 

 the examination includes more advanced and speculative subjects 

 than the Examinations for which the coefficient 15 was observed. 

 The eighth column gives the discrepancy which in each case is 

 very unlikely to occur, against the occurrence of which the odds are 

 about 100 : 1. The improbable discrepancy is by the Theory of 

 Errors equal to the probable discrepancy multiplied by 3*5 nearly. 

 In the ninth column each upper limit is formed by adding the 

 improbable discrepancy to the Honour-Line, the lower limit by 

 subtracting the same figure from the same. In the tenth column 

 the number of the successful who are uncertain is ascertained by 

 counting the number of candidates whose marks are between the 

 honour-line and the upper limit of uncertainty; the number of 

 the unsuccessful who are uncertain is found by counting the 

 number of candidates between the honour-line and the lower limit 

 of uncertainty. To form the eleventh column subtract the number 

 of the uncertain successful from the total successful ; the remainder 

 is the number of those who are " safe " in this sense that for any 

 assigned one of them the odds against his being displaced upon a 

 reexamination of his work are about 100 to 1. The number of the 

 safe divided by the number of the successful at each examination 

 is entered in the eleventh column. The laborious formation of 

 the twelfth column is described above at page 185. To form 

 the thirteenth column divide each entry in the twelfth column 

 by the corresponding entry in the fourth. The average of the 

 figures in the thirteenth column relating to the same class of 

 examination gives the proportion of the successful candidates 

 which would most probably be displaced upon a reexamination of 

 their work — most probably in the same sense as we may say that 

 the average death-rate represents the proportion of the popula- 

 tion who will most probably die in any proximate year. Thus in 

 the case of the India Civil Service we may say — or rather might 

 have said at the period to which the statistics relate, twelve years 

 ago — that the most probable proportion of displacement — the 



