546 



AMERICAN MEN OF SCIENCE 



ment; but the departure from the average 

 judgment is small, affecting less than one fifth 

 of the individuals and only to a slight degree. 

 The average deviations and probable errors 

 used are those found when all the judgments 

 are included. Two probable errors are given 

 in the table, the first obtained through the 

 error of mean square, the second by taking it 

 as directly proportional to the average devia- 

 tion. The differences are not significant, and 

 for work of this character I regard it as use- 

 less to calculate the probable errors by the 

 ordinary formula. I have published else- 

 where 10 a more technical discussion of the 

 treatment of errors or deviations of this char- 

 acter, and may return to the subject at some 

 subsequent time. The theory of errors com- 

 monly applied in the exact sciences is too 

 crude for psychology, and probably for the 

 sciences in which it is used. Progress here 

 will be blocked until there are psychologists 

 who are mathematicians or mathematicians 

 who are psychologists. 



In order to illustrate further the serial dis- 

 tribution and the probable errors, I have 

 made a diagram for the fifty psychologists. 

 The grade of each, no judgments being omit- 

 ted, is shown by the vertical mark, and the 

 length of the line indicates the probable error 

 or range within which the chances are even 

 that the true position falls. Thus the psy- 

 chologist who stands first on the list, was, like 

 the astronomer, given this position by the in- 

 dependent judgment of all. The psychologist 

 who stands second has, as shown on the dia- 

 gram, a position of 3.7 and a probable error 

 of 0.5, i. e., the position 3.7 is the most prob- 

 able, but the true position is equally likely 

 to be within the short horizontal line, between 

 3.2 and 4.2, or outside it. It must, however, 

 be remembered that the chances of the true 

 position being far outside the range of this 

 line decrease very rapidly. Over it is roughly 

 drawn the bell-shaped curve of the normal 

 probability integral. The true position is 

 along the base line covered by this curve, and 

 the chances of its being at any given point 



'Am. Journ. of Psychol., 14: 312-328, 1903. 



are proportional to the ordinate or height of 

 the curve above the base line. There is only 

 one chance in about six that the true grade 

 is above 2.7 or below 4.7, and only one chance 

 in about 150 that the true grade is above 1.7 

 or below 5.7. It will be seen from the dia- 

 gram that while the positions of the psycholo- 

 gists II., III. and IV. are the most probable, 

 the relative order is not determined with cer- 

 tainty. On the other hand, the chances are 

 some 10,000 to one that each of these psy- 

 chologists stands below I. and above V. 



It is evident that the probable errors in- 

 crease in size as we go down the list. The 



X 



10 



30 



foO 



FlO. 1. The positions and probable errors of 

 the fifty psychologists. 



curve of distribution drawn over No. XL. in- 

 dicates that the chances are even that the true 

 position falls between the grades of XXXIV. 

 and L. and that there is one chance in four 

 that he does not belong among our fifty lead- 

 ing psychologists. The increase in the size 

 of the probable errors is irregular, it being 

 more difficult to assign a position to some 

 men than to others. 



It will be noted that the psychologists fall 

 into groups, the first twenty being set off from 

 the next group, though the two groups are 

 bridged over by three cases. At this point 

 also the probable errors become almost sud- 



