t76 Prof. Edgeworth's Problems in Probabilities. 



two kinds : marks given by two examiners to several papers, 

 and marks given to the same paper by several examiners. 

 The former kind of statistics are more copious ; and in some 

 respects more valuable. For they admit of being freed from 

 a constant difference, of the nature of a " personal equation," 

 between the marking of two examiners ; which equally 

 affecting all the competitors does not disturb their order ; 

 and therefore perhaps ought not to be reckoned*. 



My method of dealing with these data may be described 

 by taking as an example the most perfect specimen which I 

 obtained, namely the marks given by two examiners to 400 

 pieces of English composition. First I took the difference 

 between the two marks given to each paper ; then squared 

 those differences, found the Arithmetical Mean of those 

 squares, and took the square root of that Mean Square as the 

 Mean Deviation in a sense analogous to Gauss's use of Mean 

 Error. The peculiar propriety of this coefficient as a measure 

 of discrepancy is that it not only represents, as well as other 

 sorts of average error, the deviation between marks in any 

 particular subject ; but also leads to the coefficient of the 

 Probability-curve which measures the deviation between sums 

 of marks. Thus the Mean Deviation for the 400 pairs of 

 marks in English composition proved to be 67, which co- 

 efficient not only gives a general idea of the discrepancy to 

 be expected between any two marks, but also yields a 

 precise system of measures for the discrepancy between the 

 sum of several marks assigned by two examiners to the same 

 pieces of work in pari materia. That discrepancy would 

 fluctuate according to a Probability-curve whose modulus is 

 \l% x 10 x 67, or whose probable error is "674 . . . x v^lO x 67. 



For our purpose it is generally convenient to express the 

 Mean Deviation as ,a percentage of the mean mark for a 

 whole set of papers. Thus in the case before us the average 

 of the 800 marks was 227 ; and accordingly the Mean Devia- 

 tion per cent, in round numbers 30. 



This result requires to be corrected for a certain " personal 

 equation." The constant difference between the two sets of 

 marks is about 20, nearly ten per cent, of the average mark. 

 The square of this constant difference is to be subtracted from 

 the uncorrected Mean Square ; of which 67 was the square 

 root. The corrected Mean Deviation is 64 ; expressed as a 

 percentage of the average mark, 28 nearly. 



The worth of this result may be appreciated from the state- 



* See on this point the companion paper in the 'Journal of the 

 Statistical Society ' for 1890. 



