On the Mathematical Theory of Errors of Judgment. 369 



XXVI. " On the Eelation between the Electrical Resistances of Pure 

 Metals and their Molecular Constants." By W. Williams. 

 Communicated by Professor Andrew Gray, F.E.S. 



The Society adjourned over the Long Vacation to Thursday, 

 November 21, 1901. 



" On the Mathematical Theory of Errors of Judgment, with 

 Special Eeference to the Personal Equation." By Karl 

 Pearson, F.R.S., University College, London. Eeceivecl April 

 23,— Eead June 20, 1901. 



(Abstract.) 



In 1896 I, with Dr. Alice Lee and Mr. G. A. Yule, made a series 

 of experiments on the bisection of lines at sight. The object of these 

 experiments was to test a development of the current theory of errors 

 of observation, by which it seemed possible to me to determine the 

 absolute steadiness of judgment of any individual by comparing the 

 relative observations of three (instead of as usual two) observers. As 

 a rule the absolute error of the observer is unknown and unknowable, 

 and I was seeking for a quantitative test of steadiness in judgment to 

 be based on relative judgments. If o- 01 be the standard deviation 

 of the absolute judgments of the first observer, qr 12 , cr 23 , or 31 the 

 standard deviations of the relative judgments of the first and second, 

 the second and third, and the third and first observers respectively, 

 then 



°-oi 2 = i(o2i 2 + <r l8 a-oW0 W 



on the basis of the current theory of errors. Thus it seemed possible 

 to determine absolute steadiness of judgment from the standard devia- 

 tions of relative judgments, which are all that the physicist or astro- 

 nomer can usually make, provided three observers and not two were 

 compared. 



To my great surprise I found results such as (i) were not even 

 approximately true, and that they failed to hold because the judg- 

 ments of the observers were substantially correlated. It did not occur 

 to me at first that judgments made as to the midpoints of lines by 

 experimenters, in the same room it is true, ,but not necessarily bisect- 

 ing the same line at the same instant, could be psychologically corre- 

 lated, and I looked about for a source of correlation in the treatment 

 of the data. We had taken 500 lines of different lengths and bisected 

 them at sight ; assuming that the error would be more or less propor- 

 tional to the length of the line, I had adopted the deviation from the 



