Determination of the Modulus of Errors. 501 



which require, no remedy. On the one hand are the problems 

 in which it is required to determine the constant in question, 

 not from an ample registry of observations*, but from a few 

 samples. On the other hand are problems which follow the 

 analogy of games-of-chance ; where the constant is predeter- 

 mined, and does not require to be elicited from a register of 

 observations. The intermediate case, where the modulus is 

 to be determined empirically and the material for determining 

 it is copious, is the most frequent in practical statistics. Sup- 

 pose, for instance, it has been observed that the death-rate in 

 a certain occupation exceeds by a certain amount the rate in 

 other occupations. To determine whether or not that excess 

 is accidental, we require to know the modulus of the proba- 

 bility-curve under which such death-rates range. Or, if it 

 is observed that the mean height of a certain class f, the Royal 

 Society for instance, exceeds the general mean, to complete 

 the argument based on that excess it may be requisite to know 

 what degree of excess would be likely or unlikely to be pre- 

 sented by the mean height of a batch of men taken at random 

 from the general population. The latter example may be con- 

 trasted with the former in this respect, that the quantity with 

 which it deals — linear dimension — is continuous, not discrete. 

 It follows that, in examples of the latter type, the data for the 

 calculation of our constant are apt not to possess the perfec- 

 tion of which they are theoretically susceptible. We are 

 usually given the number of men corresponding to each 

 degree of height ; but the scale is not finely graduated. For 

 example, in the statistics just referred to the unit is an inch. 

 In the copious statistics given by Dr. Baxter in the Report 

 of the Sanitary Commission of the United States, the unit is 

 tivo inches. To treat such finite differences as differentials may 

 introduce error. 



The cases which present this difficulty may be divided, 

 according as we have, or have not, preliminary knowledge of 

 the form or family of the curve whose radius of gyration is 

 required. In the former case we may proceed by first redu- 

 cing our imperfect data to the known form, and then calcu- 

 lating for the curve thus determined the radius of gyration. 

 The second part of the operation is particularly easy in the 

 case of most frequent occurrence, namely where the presumed 



* In this case the required constant must be elicited by Inverse Pro- 

 bability, as explained in my paper on Observations and Statistics 

 (Cambridge Philosophical Journal) under the heading 7 



t Cf. Report of the Anthropometrical Commission of the British 

 Association. 



Phil, Mag. S. 5. Vol. 21. No. 133. June 1886, 2 N 



