Prof. Edge worth's Problems in Probabilities. 181 



where, the candidates and prizes being indefinitely nume- 

 rous, the Honour-line may be regarded as fixed, our Bxyz. 

 It will be observed that the measure of discrepancy for Bxyz 

 is less than the corresponding measure for the extreme case 

 of Bxyz in the ratio 1 : \''2. We have thus a rough measure 

 of the inaccuracy which we commit in treating intermediate 

 cases according to the rule proper to either extreme. 



Bxyz. This relation of this case to the preceding is like 

 the relation of Bxyz to Blcyz; which has already been con- 

 sidered. The coefficient of discrepancy which is proper to 

 Bxyz should be multiplied by \/2 for Bxyz. 



Bxyz. This case differs from the preceding in that the 

 number of the papers is small. At this stage therefore the 

 Probability-curve which has hitherto illumined our course 

 disappears. A certain twilight may, however, be derived 

 from that source of illumination. Take the extreme case of 

 two candidates examined in one subject for a prize. What is 

 the probability that the award of one examiner would be 

 reversed by an equally competent examiner? As we saw 

 under a preceding head, the answer turns upon the variation 

 with the change of examiners in the difference between the 

 marks of the two candidates. Such a difference between 

 differences of marks will in general fluctuate according to the 

 same law as a sum of four marks taken at random from under 

 the facility-curve, according to which by our postulate the 

 marks of different examiners fluctuate. But a sum of four 

 observations taken from under any facility-curve will in 

 general fluctuate according to a law which is getting on for 

 a Probability-curve, unless indeed the given facility-curve be 

 exceedingly abnormal. But so far is the facility-curve with 

 which we have to deal from being exceedingly abnormal, 

 that it is presumably getting on for a Probability-curve. 

 Accordingly the rule for Bxyz may pretty safely be extended 

 to Bxyz ; especially where we have specific experience that 

 the facility-curve in question does not violently rebel against 

 the normal law of error — experience which 1 have obtained 

 with respect to several subjects. For example, let a prize be 

 given to the one of two candidates who obtains the higher 

 mark for a piece of Latin Prose, of about the same calibre as 

 the Composition at the India Civil Service or Cambridge 

 Classical Tripos Examinations. Even if the successful candi- 

 date exceeds his rival by fifty per cent, (of the mean between 

 the two marks), there is some probability, say one in a hun- 

 dred, that the verdict would be reversed by another equally 

 competent examiner. 



