On the Mathematical Theory of Errors of Judgment. 371 



The current theory of the personal equation thus appears to need 

 modification, and we require for the true consideration of relative 

 judgments not only a knowledge of the variability of observers, but 

 also of their correlation in judgment as necessary supplements to the 

 simple personal equation. 



Having obtained from our data twelve series of errors of observation 

 considerably longer than those often or even exceptionally dealt with 

 by observers, we had a good opportunity for testing the applicability of 

 the current theory of errors, in particular the fitness of the Gaussian 

 curve 



y = y e-*w> 



to describe the frequency of errors of observation. In a considerable 

 proportion of the cases this curve was found to be quite inapplicable. 

 Errors in excess and defect of equal magnitude were not equally 

 frequent ; skewness of distribution, sensible deviation of the mode from 

 the mean, "crowding round the mean," even in the case of passable 

 symmetry, all existed to such an extent as to make the odds against 

 the error distributions being random samples from material following 

 the Gaussian law of distribution enormous. It is clear that deviation 

 of the mode from the mean, and the independence of at least the first 

 four error moments, must be features of any theory which endeavours 

 to describe the frequency of errors of observation or of judgment 

 within the limits allowable by the theory of random sampling. The 

 results reached will serve to still further emphasise the conclusions I 

 have before expressed : 



(a.) That the current theory of errors has been based too exclusively 

 on mathematical axioms, and not tested sufficiently at each stage by 

 comparison with actual observations or experiments. 



(b.) That the authority of great names — Gauss, Laplace, Poisson — 

 has given it an almost sacrosanct character, so that we find it in current 

 use by physicists, astronomers, and writers on the kinetic theory of 

 gases, often without a question as to its fitness to represent all sorts of 

 observations (and even insensible phenomena !) with a high degree of 

 accuracy. 



(c.) That the fundamental requisites of an extended theory are that 

 it must — 



(i.) Start from the three basal axioms of the Gaussian theory and 

 enlarge and widen them. 



(ii.) Provide a systematic method of fitting theoretical frequencies 

 to observed distributions with (a) as few constants as possible, these 

 constants easily determinable and closely related to the physical charac- 

 ters of the distribution, and 



(iii.) When improbable isolated observations are rejected, give theo- 

 retical frequencies not differing from the observed frequencies by more 

 than the probable deviations due to random sampling. 



