270 Mr. F. Y. Edgeworth on 



Curve whose modulus is — j= — , P being the Greatest Ordinate 



of the common facility-curve. The criterion whether the 

 Median or Arithmetic Mean is the better reduction*, is pre- 

 sumably the character of the correlated Probability- Curve f. 

 The reduction to which corresponds the smaller Modulus is 

 presumably the better ; since thus we obtain a smaller " pro- 

 bable " error, and, what is often more important, a smaller 

 improbable, or, as Mr. Merriman proposes to call it, " huge " 

 error. 



Which of the reductions will have the smaller Modulus 

 will depend upon the character of our facility-curve. For 

 Probability-Curves, and presumably functions in their neigh- 

 bourhood, it is shown by LaplaceJ that the Arithmetic Mean 

 has the advantage. But for curves whose head reaches high, 

 while their extremities stretch out far, the Median has the 

 advantage§. 



Now the grouping of Discordant Observations is apt to 

 assume this form. Accordingly the Median is proposed as 

 the Mean proper to this class of observations. If we have 

 been deceived by the appearance of Discordance || (as in Gen. 

 Colby's case, cited by Airy), and the facility-curve was really 

 a normal Probability-Curve, yet we shall have lost little by 

 taking the Median instead of the Arithmetic Mean. For the 

 errorlf of the former is of the same order as (only 1*3 greater 

 than) the error of the latter. And, if the observations are 

 really discordant, the derangement due to the larger devia- 

 tions will not be serious, as it is for the Arithmetic Mean**. 



In illustration of the former proposition, let us take forty 

 observations made by Bessel (on Saturn's ring), which Prof. 

 Chauvenet has cited ff as presumably of the normal type. The 

 Median is found by arranging the observations in the order 

 of magnitude and taking a mean between the twentieth and 

 twenty-first J J. They are both +'01, the Arithmetic Mean 



* Dr. Venn's criterion is much the same. 



t Laplace, loc. cit. Cp. Camb. Phil. Trans. 1885, pp. 168-9. 



\ Loc. cit, 



§ Camb. Phil Trans, p. 168. 



|| See the paper on Discordant Observations in the April number of this 

 Journal. 



51 Laplace, loc. cit. 



** This advantage of the Median over the Arithmetic Mean has been 

 noticed by Fechner, Mr. Galton, and others. 



tt ' Astronomy,' vol. i. p. 495. 



X% There is a little indeterminateness here (the number of observations 

 being even) of not much practical importance when the number is large. 



