SCIENCE 



ferent men do not have the same weight. The 

 ten \vto receive the most votes receive nearly 

 all the votes, and in the ca.ses of the few who 

 do not vote for them poor judgment or an error 

 in checking is indicated. Some votes mean that 

 a psychologist stands first or near the top of the 

 list, whereas others mean that he is barely 

 included. Consequently the order and the 

 piobable errors in the ease of such a vote for 

 one half of the group do not have great validity 

 for the upper part of the distribution. The 

 order is obtained in a satisfactory manner by 

 the double vote, but this introduces further 

 complications in the probable error. In all 

 cases of votes, we have asymmeti-ical distribu- 

 tions and skew curves. The quantitative rela- 

 tions should be worked out in the first instance 

 for less complicated material than that with 

 which we are here concerned. 



In nearly all cases in which probable errors 

 have been applied to psychological data, the 

 determinations are more exact than common 

 sense would presuppose. Thus the writer 

 found* that in grading traits of character by 

 ten individuals on a scale of 100, a position was 

 assigned with probable errors varying from 

 4.6 for physical health and cheerfulness to 3 

 for originality and efficiency. All other traits, 

 such as energy, courage, judgment and integ- 

 rity, were assigned positions with intermediate 

 probable errors, the average being 4, which is 

 nearly the same as the probable error of posi- 

 tion as determined by 80 votes of the psycholo- 

 gists near the middle of the 100 in order of 

 merit. 



The comparatively small proba1)le errors ap- 

 pear to be due to the fact that there ai'e con- 

 stanlt errors which affect the whole group. The 

 psychologists who vote are subject to the same 

 kind of influences, not making in fact inde- 

 pendent judgments, but being influenced as a 

 group by the knoAvledge of what others think 

 and hy all sorts of conditions, conventions and 

 restrictions. If a similar vote were taken ten 

 years hence the work of the same psychologists 

 would be viewed from new standpoints and the 

 pesitions would change to a much greater de- 

 gree than the j)io)iable errors warrant. "Con- 



* Address of the president of the Amcrii-aii 

 fcjocietj of Naturalists, Science, April 10, 1903. 



stant" errors are in fact more inconstant and 

 variable than "variable" errors. 



In the case of a vote (as in any series of 

 measurements) there are two factors entering 

 into the probable error, one dependent on the 

 quantitative conditions prescribed in advance, 

 the other on the behavior of the individuals. 

 The former may be called the deductive prob- 

 able error and when the latter is determined l)y 

 experiment and added to it the whole is the 

 inductive or actual probable error. Thus, if 

 from an indefinitely large number of balls 

 equally distributed between black and Avhite, 

 some are drawn, the most frequent distribution 

 will be an equal number of black and white, 

 but the average departure from equality will 

 increase as the square root of the number 

 drawn and the ratio of departure from equality 

 will decrease as the square root of the number. 



If large numbers of white and black balls are 

 distributed in the ratio of 33 white to 47 black, 

 and we draw 80 balls, the most probable num- 

 ber of white balls will be 33. The standard 

 deviation from 33 in a large number of draws 

 will be 4.40, and the quartile deviation or prob- 

 able error will he 2.97; that is, in one case out 

 of four there will be more than three white 

 balls. The psychologist at the bottom of the 

 fifty received 33 votes out of a possible 80. If 

 an indefinitely large number of psychologists 

 were distributed in this ratio the deductive 

 probable error or error of sampling would be 

 2.97. The actual prc^bable error, namely, 3.63, 

 is composed of (the square root of th sum of 

 the squares of the two) bhis deductive probable 

 error and an error or deviation due to the 

 groupings of the psychologists into different 

 "species" with different points of view. The 

 psychologist who stood XIL had a probable 

 error of 3.1. The deductive probable errors 

 are approximately the same for the two indi- 

 viduals, but it is more difficult to form a judg- 

 ment regarding No. L than regarding No. XIL.^ 



'' In these cases the actuiil and the deductive 

 probable errors have probable errors of the order 

 of magnitude of the differences between them, 

 and these differences have only moderate validity. 

 The writer has purposely "not minded his p's 

 ami his q's," for it seems that equations are not 

 becoming to one who is not a mathematician. 



