172 Prof. Edgeworth's Problems in Probabilities. 



rather abrupt from objective quantities observed by the senses 

 to degrees of intellectual quality estimated by the judgment, 

 I have confirmed the analogy by trying an experiment in an 

 intermediate case, where the quantity to be determined is 

 objective, but the operation by which it is determined is an 

 estimate rather than a simple observation. Of this sort are 

 the answers which I have obtained by asking a number of 

 persons separately : " What is my weight ? " Having col- 

 lected 96 estimates made in reply to this question, I find that 

 they are constant to a definite law of frequency. That law 

 is exhibited in the annexed table or scheme ; in which the 

 ordinary numerals denote pounds above nine stone (thus 25 

 means 10 st. 11 lb.), and the Roman numerals above denote 

 the number of estimates which are entered in each of the 

 spaces bounded by the ordinary numerals (thus there were 

 four estimates below 9 st. 10 lb., and nineteen between that 

 figure and 10 st. 6 lb.) *. The verification of our axiom is 



IV. XIX. XXIV. XXVI. XIX. IV. 



10. 20. 25. 30. 40. 



II. XIV. X. XV. VI. t 



tl. VII. XIII. XII. XII. in. 



shown by the lower rows of Roman numerals which respec- 

 tively designate the distribution of the first 48 answers which 

 I received, and of the second batch. I add the following 

 verifications. Regarding as an error the deviation of any 

 estimate from the average of all J, namely 25, I find that the 

 average of the errors in defect is 7*2 ; the average of the 

 errors in excess is 6*8. Now split up the forty-seven errors in 

 defect into two batches as nearly as may be ; the average of 

 the first twenty-four — first in the order of arrival — is 7*5; of 

 the remaining twenty-three, is 6'S. So the average of the 

 first twenty-five errors in excess is 5-6, of the remaining 

 twenty-four 8. 



Even the batches of sixteen show considerable steadiness. 

 The following table exhibits this constancy. The first column 

 designates the position of each batch of sixteen in the 

 accidental order in which it was received and entered. Thus 



* Where a number of estimates coincided at one of the boundaries, 

 e. g. 20, 1 gave half to one compartment, e. g. IV.-XIX., half to the other, 

 XIX.-XXIV. Where the number was odd I gave the benefit to the com- 

 partment nearer to the centre. 



t The apparent anomaly that the whole of certain compartments con- 

 tains more or less than the sum of the parts is explained by the preceding 

 note. 



% Average in this paper stands for Arithmetic Mean. 



