700 



SCIENCE. 



[N. S. Vol. XXIV. No. 622. 



Here we find complete agreement that I. 

 is our leading astronomer. He has been se- 

 lected as such by nine competent judges from 

 the 160 astronomers of the country.^ The 

 probability that this is due to chance is en- 

 tirely negligible. II. stands next in scientific 

 merit. He is placed second by four of the 

 observers, third by two, fourth by three and 

 ninth by one. The conditions are similar to 

 observations in the exact sciences. The av- 

 erage position or grade is 3.5, and the prob- 

 able error of this position is 0.45, i. e., the 

 chances are even that this grade is correct 

 within one half of a unit. The grade of the 

 astronomer who stands third is 4.8, and that 

 of the astronomer who stands fourth is 5.5. 

 There is consequently one chance in about 

 fifty that II. deserves a grade as low as that 

 of III., and one in about one thousand that 

 he deserves a grade as low as that of IV. The 

 order thus has a high degree of validity, and 

 this has itself been measured. As we go 

 further down the list, the probable errors tend 

 to increase, the order is less certain, and the 

 difference in merit between a man and his 

 neighbor on the list is less. The variations 

 in the sizes of the probable errors are, as a 

 rule, significant. When the error is small 

 the work of the man is such that it can be 

 judged with accuracy; when it is larger it is 

 because the work is more difficult to estimate. 



The probable errors depend on the assump- 

 tion that the individual deviations follow the 

 exponential law, and they do so in sufficient 

 measure for the purposes in view. For those 

 near the top of the list, the distribution of 

 errors is ' skewed ' in the negative direction, 

 that is, there are relatively more large nega- 

 tive than positive errors. Thus in the table 

 there are four judgments marked with a star, 

 the deviation of each of which is more than 

 three times the average deviation, and these 

 observations would be omitted by an ap- 

 proximate application of Chauvenet's cri- 

 terion. If these four observations are omit- 

 ted, the grades of the ten astronomers are 



^ In three cases where a question mark appears 

 the astronomer did not give a position to himself. 

 In one case the name was not included among the 

 slips. 



those given in the second line of averages. 

 The omitted judgments are not extremely di- 

 vergent, barely exceeding the limits set by 

 Chauvenet's criterion, and I do not regard 

 them as invalid. Indeed, I believe that in 

 view of the presence of systematic errors in 

 these estimates the chance that they represent 

 correct values is greater than that assigned 

 by a. strict application of the theory of proba- 

 bilities. But the incidence of an extreme 

 judgment might in special cases do injustice 

 to an individual, and in the order used Chau- 

 venet's criterion has been applied." This 

 means that a compromise has been adopted 

 between the median and the average judg- 

 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^ 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 



^ Among the some 15,000 observations under 

 consideration several variations might be expected 

 to occur in a normal distribution as much as six 

 times as large as the probable error, and among 

 the 1,500 or more individuals, several might be 

 expected to deserve positions departing consider- 

 ably from those assigned. But assuming that we 

 have 'normal errors' to deal with, there is no 

 reason why the particular individuals on whom 

 the divergent errors fall should receive them 

 rather than any other individuals. Such errors 

 should apparently be distributed among all the 

 individuals. Similar conditions must occur in 

 the case of errors of observation in the exact 

 sciences, but so far as I am aware their signifi- 

 cance has not been considered. 



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



