TEANSACTIONS OF SECTION F. 763 



give a practical imperial federation, and point out a practical system of co-operation 

 between Capital and Labour which might ultimately confer safety and comfort 

 upon civilised society. 



5. On the Statistics of Examination} 

 By Professor F. T. Edgeworth, M.A., F.8.S. 



The object of this paper is to estimate the error incident to a large aggregate of 

 marks at a competitive examination in several subjects. Error is detined as devia- 

 tion from that Mean towards which the average marking of numerous competent 

 critics -would tend. . 



Among the sources of error are : I. The fact that there is a minimum sensibile^ 

 in our perception of degree of merit. II. The idiosyncrasy or personal equation of 

 the examiner affecting (1) particular answers, (2) the scale of marks in any subject. 

 III. The negligence of examiners. 



The extent of the several errors is estimated by means of statistical data. Com- 

 pounding the items according to the rules of the Calculus, the writer finds a 

 ' probable error ' of 2 per cent, (corresponding to a possible error of more than 

 6 per cent.) in the aggregate of marks according to which candidates are classed at 

 an examination of a kind now prevalent. 



That is the error in appreciating the work done at the examination. A further 

 error is incurred in taking that work as a sample of the candidate's real proficiency. 

 Upon a certain mechanical hvpothesis, a part of this latter error can be estimated by 

 the Calculus of Probabilities. Adding this error to the previous accoiint, we may 

 infer that the number of candidates displaced— that is, not obtaining a prize 

 when they ought, or obtaining one when they ought not — is likely to be about 30 

 out of a total of 1,000 examined. 



The following recommendations are based on statistical grounds : (I) The order 

 of merit withm the Honours Class should be abandoned whenever the difference 

 between the marks of the discriminated candidates is less than the 'probable error' 

 incident to those figures. (2) It might be expedient that the examiners should at 

 first scrutinise only half the answers of each candidate, wherever the purpose of the 

 examination would be subserved by obtaining a rough preliminary indication of the 

 order of merit. (3) In order that the importance assigned to each subject may be 

 free from accidental oscillation, it would be allowable to assume that the average 

 proficiency of a large number of candidates in any subject is, from year to year, con- 

 stant (just as the average height is) ; and, therefore, that the mean of the marks in 

 that subject should be the same every year. If the average of the marks given to a 

 number of candidates by any examiner differ from the figure which may have been 

 prescribed for the mean, those marks should be reduced to the normal scale. The 

 practice of basing the standard upon the answering of the most proficient can- 

 didate is objectionable. (4) For a certain purpose it might be theoretically 

 desirable to combine the marks in different subjects, not by simple addition, but by 

 taking their geometrical mean. 



SATUBBAY, SEPTEMBER 8. 

 The following Papers were read : — 

 1. The Revenue System of the Vnited States.^ By Albert Shaw, Ph.B. 



The United States enters upon the current fiscal year with an interest-bearing 

 debt reduced below ^^1,000,000,000. In 1865 the bonded debt was approximately 

 ^2,400,000,000. Sixty per cent, of the principal has been wiped out, and 75 per 



■ Published in externa in the Journal of the Royal Statistical Society, Sept. 1888. 

 * Published in the Contemporary Review, Nov. 1888. 



