Prof. Edge worth's Problems in Probabilities. 185 



In this Table the first column denotes degrees of probable 

 error corresponding to intervals of marks designated in 

 columns 2 and 4. Columns 3 and 5 give the number of can- 

 didates successful and unsuccessful whose marks fall in those 

 intervals ; column 6 the total of those numbers. Column 7 



2 A 

 contains the values of half of the integral — = j e~ x>i dx for 



x = Probable Error* x '1, "2, &c. — 1*8, 2'2 respectively (values 

 obtained from the fourth table appended to DeMorgan's 

 'Calculus of Probabilities'). Column 8 gives the difference 

 between '5 and each of these values, the corresponding inte- 

 gral between limits go and x ; which represents the proba- 

 bility of displacement for candidates in the corresponding 

 compartment. Column 9, the product of column 6 and 

 column 8, gives the most probable of number of those who 

 would be displaced, for each degree or interval. The sum of 

 these numbers is the most probable number displaced, out of 

 all the candidates. I take the half of this number as the most 

 probable number of successful candidates who would be dis- 

 placed on re-examination. 



In conclusion I submit a Table containing answers to 

 Problems 2 and 3 for certain cases which seem to me fairly 

 typical of the various statistics which I have inspected. 



In this Table the first column designates a service to which 

 appointment is made by competitive examination. The second 

 column contains references to the Reports of the Civil Service 

 Commissioners, in which are published the marks given at public 

 examinations. The Reports referred to are in the 22nd vulume 

 of the Parliamentary papers for 1875 and for 1876. The third 

 column gives the number of candidates at each of the examinations 

 referred to in this table. The fourth column gives the correspond- 

 ing numbers of successful candidates. The fifth column contains 

 the mark of the lowest successful candidate at each of the 

 examinations, or of the highest unsuccessful, or some intermediate 

 number (figures differing from each other by quantities which may be 

 neglected). The sixth column contains the aggregate marks which 

 occupy the halfway position in the order of merit at each ex- 

 amination. Thus at the second examination (referred to in the 

 second row of the table) there being 171 candidates, the aggregate 

 mark which is 86th in the order of merit is 1601 ; in round 

 numbers 1600. At the first examination, the number of candidates 

 being even, viz. 150, the Median is intermediate between the 75th 

 mark, which is (in the descending order of merit) 1601, and the 



* See footnote, p. 183. 



