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LXIX. On the Determination of the Modulus of Errors. 

 By F. Y. Edgewobth, M.A., Lecturer on Logic, King's 

 College, London*. 



EOR most of the applications of the Law of Errors f there 

 is required the determination of a constant which may 

 be described as the mean-square-of-error upon the supposition 

 that the centre-of-gravity is the point of null error — in short 

 the radius of gyration — for a certain plane curve. This curve 

 is the facility-curve under which a given observation ranges J: 

 the curve which would be generated if we registered an inde- 

 finite number of observations made under unaltered conditions. 

 Take, for example, Laplace's proof§ that the difference be- 

 tween the mean of 400 barometrical observations taken at 

 4 p.m. and the mean of observations || taken at 9 a.m. is not 

 accidental, but indicative of a cause. The reasoning turns 

 upon the incident that the mean of 400 observations may be 

 regarded as ranging under a probability-curve, whatever the 

 curve under which the individual observations range. In 

 Laplace's example it is supposed to be given that an observa- 

 tion is as likely to occur at one point as another of a line 

 measuring 8 millimetres, but cannot occur outside that range. 

 The facility-curve then for each of the individual observations 

 is a rectangle of base 8 and of height £ inch. The mean- 



1 R. 



square-of-error for this facility-locus being -^-, we know that 



the modulus-squared of the probability-curve under which the 



-is* -i 



mean of 400 observations ranges is 2 x — -^400= ^=- c . 



Whence it is deducible that the observed difference, namely 

 one millimetre, being about six times the modulus, is very 

 unlikely to have occurred by mere chance. 



The object of this paper is to soften the difficulty which the 

 calculation of this constant is apt to present. The cases which 

 it is proposed to treat are intermediate between the least and 

 the most perfect data : between those which admit, and those 



* Communicated by the Author. 



t For some account of these see the references given at the beginning 

 of my paper in the April Number of this Journal. 



% Mr. Todhunter's y=f(z). (Hist, of Prob. art. 1001.) 



§ Theorie Analytique, Book II. ch. v. 



|| It will be found, I think, that Laplace's reasoning assumes the morn- 

 ing mean (which also rests upon 400 observations) as a fixed point. It 

 might have appeared more natural to find the modulus for the difference 

 between the two means on the supposition that neither was perfectly 

 determined by 400 observations. 



