AMERICAN MEN OP SCIENCE 



545 



TABLE I. THE OBDEB ASSIGNED TO TEN ASTEONOMEBS BY TEN OBSEBVEBS 



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- 

 36 



terion. If these four observations are omit- 

 ted, the grades of the ten astronomers are 

 those given in the second line of averages. 

 The omitted judgments are not extremely 

 divergent, 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- 



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. 



