November 30, 1906.] 



SCIENCE. 



701 



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 



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 

 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 



IX^ 



X-. 



^'"^ 



10 ZO 30 ifO SO GO 



Fig. 1. The positions and probable errors of the fifty psychologists. 



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 



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 

 curve of distribution drawn over No. XL. in- 

 dicates that the chances are even that the true 



