Discordant Observations. 371 



This gives for the required limit about 15. According, then, 

 to II. (1) (/3), any observation greater than 60, or less than 30, 

 is more likely than not to be a mistake in the sense of not 

 belonging to the same law of frequency as the observations 

 within those limits. But why on that ground should the 

 discordant observation be rejected ? Suppose there were not 

 merely a bare preponderance of probability, but an actual 

 certainty, that the suspected observation belonged to a different 

 category in respect of precision from its neighbours, the best 

 course certainly would be if possible (as Mr. Glaisher in his 

 paper " On the Rejection of Discordant Observations " sug- 

 gests) to retain the observation affected with an inferior weight. 

 But if we have only the alternative of rejecting or retaining 

 whole, it is a very delicate question whether retention or re- 

 jection would be in the long run better. There is not here 

 the presumption against retention which arises when, as in 

 II. (1), the discordant observation is large and rare ; so that, 

 if it is a mistake, it is likely to be a serious and an uncom- 

 pensated one. However, Prof. Chauvenet's method may 

 quite possibly be better than the No-method of Sir Gr. Airy. 

 Much would turn upon the purpose of the calculator — whether 

 he aimed at being most frequently right* or least seriously 

 wrong. The same may be said with reference to hypo- 

 thesis (7). 



There is a further difficulty attaching particularly to this 

 species of Method II. In its precise determination of a limit, 

 it takes for granted that the probability-curve to which we 

 refer the discordant observation is accurately determined. 

 But, when the number of observations is small, this is far 

 from being the case. Neither of the parameters of the curve, 

 neither the Mean, nor the Modulus, can be safely regarded as 



c 

 accurate. The 6i probable error " of the Mean is *477 —7= 

 1 v n, 



where c is the Modulus. The probable error of the Modulus 



is conjectured to be not inconsiderable from the fact that, if 



we took m observations at random, squared each of them and 



formed the Mean-square-of-error, the " probable error " of that 



c 2 

 Mean-square-of-error would be *477 -—r^ f. This, however, is 



not the most accurate expression for the probable error of the 

 Modulus-squared as inferred % from any given n observations. 



* See the remarks above, p. 369. 



t Todhunter, art. 1003 (where there is no necessity to take the origin 

 at one of the extremities of the curve). 



} I allude here to delicate distiuctions between genuine Inverse Pro- 

 bability and other processes, which I have elsewhere endeavoured to 

 draw, Canib. Phil. Trans. 1885. 



2C2 



