368 Mr. F. Y. Edgeworth on 



species : the criteria of Prof. Stone*, Prof. Chauvenetf , and 

 Prof. PeirceJ. 



II. (1) Prof. Stone's method is to reject an observation 

 when it is more likely to have been a mistake than an error 

 of observation of the same type as the others. In deter- 

 mining this probability he takes account of the a priori 



probability of a mistake. He puts for that probability -, 



admitting that n cannot be determined precisely. The use of 

 undetermined constants like this is, I think, quite legitimate?, 

 and, indeed, indispensable in the calculation of probabilities. 

 This being recognized, Prof. Stone's method may be justified 

 upon almost any hypothesis. Hypothesis (a) presents two 

 cases : where the discordant observation exceeds that limit of 

 errors proper which is known beforehand, and where that 

 limit is not exceeded. For example, in the instance || given 

 above — where 45 is the Mean, and the Moduluslf is about 13 — 

 the discordant observation might be either above 100 (e.g. 110) 

 or below it (e. g. 84) . Now let us suppose that the a priori 

 probability of a mistake is not infinitesimal, but say of the 

 order yq'qq. Since the deviation of 110 from the Mean is 

 about five times the Modulus, the probability of this deviation 

 occurring under the typical law of error is nearly a millionth. 

 This observation is therefore rejected by Method II. (1), which 

 so far agrees with Method I. Again, the probability of 84 

 being an accidental deviation is less than a forty- thousandth; 

 84—45 being about three times the Modulus. Therefore 84 also 

 is rejected by the criterion. And we thus lose an observation 

 which is by hypothesis (a) a good one. But this loss occurs 

 very rarely. And the observation thrown away is, to say the 

 least, not** a particularly good one, though doubtless it may 

 happen that it is particularly wanted — as in the case of Gen. 

 Colby, adduced ft by Sir G. Airy. 



II. (1) (/3) The second hypothesis is that to which Prof. 

 Stone's criterion is specially adapted. Upon this hypothesis, 

 84 may be a mistake. In rejecting such discordant observa- 

 tions, we may indeed lose some good observations, especially if 



* Month. Not. Astronom. Soc. Lond. vol. xxviii, pp. 165-168. 



t ' Astronomy/ Appendix, Art. 60. \ Ibid. Art. 57. 



§ See my paper on a priori Probabilities, in Phil. Mag. Sept. 1884 ; also 

 " Philosophy of Chance,* Mind, 1884, and Camb. Phil. Trans. 1885, 

 pp. 148 et seq. || Page 365. 



^] Vl65, exactly. As determined empirically by me from the mean- 

 square-of-error of 280 observations (i. e. sums of 10 digits), the Modulus 

 was a/160. 



** See the remark made under II. (2) (/3). 



ft Gould's Astronom. Journ. vol. iv. p. 138. 



