546 



SCIENCE 



[Vol. LVI, Xo. 1454 



determinations are more exact than common 

 sense ■would presupisose. Thus the writer 

 found' that in grading traits of character by 

 ten individuals on a seals of 100, a position was 

 assigned ivitli probable errors varying from 

 4.6 for physical health and cheerfulness to 3 

 for originality and eiScieney. All other traits, 

 such as energy, courage, judgment and integ- 

 i-ify, were assigned positions with intermediate 

 probable errors, the average being 4, Avhieh 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 probable errors ap- 

 pear to be due to the fact that there are con- 

 stant 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 knowledge of what others think 

 and by all sonts of conditions, conventions and 

 restiietions. If a similar vote were taken ten 

 years hence the work of the same psychologists 

 would be viewed from new standpoints and the 

 230sitions would change to a much greater de- 

 gree than the probable errors warrant. "Con- 

 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 proba])le 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 by 

 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 white, 

 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 lilack 'balls are 



■* Address of the president of the American 

 Society of Xaturalists, Science, April 10, 1903. 



distributed in the ratio of 33 white to 47 black, 

 and we draw 80 balls, the most probable raun- 

 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- 

 al)le error will 'be 2.97; that is, in one case out 

 of four there will be more than three white 

 balls. The psj'chologist 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 proibaible error, namely, 3.63, 

 is composed of (the square root of the smn of 

 the squares of the two) t'his deductive probable 

 error and an error or deviation due to the 

 groupings of the ps5'-chologists into different 

 "species" with different jjoints of view. The 

 psychologist who stood XIL had a probable 

 error of 3.1. The deductive protoa'ble 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.^ 

 The situation may be illustrated by an in- 

 stance of general importance. Death rates, 

 birth rates and marriage rates are continually 

 used, but always without probable errors. 

 Thus, for example, the Bureau of the Census 

 issues weekly a bulletin that contains the death 

 rates of the leading cities of the United States, 

 but the figures have no meaning because one 

 does not know whether the different rates are 

 due to chance fluctuations with a limited popu- 

 lation or to causes such as a large proportion 

 of infants or an epidemic of influenza." If the 



•5 In these cases the actual and the deductive 

 probable errors have probable errors of the order 

 of magnitude of tl'.e differences between tliem, 

 and tliese differences have only moderate validity. 

 The writer lias purposely "not minded his p 's 

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

 becoming to one who is not a mathematician. 



f' In the last report received (for the week end- 

 ing September 2, 1922) the death rate of New 

 Haven is given as 5.8 and of Houston as 13.9. In 

 the same week a year ag'O the death rate of New 

 Haven was 10 and of Houston 7.6. Without 

 probable errors these figures give no useful 

 information in regard to the conditions in the two 

 cities. 



