372 Mr. F. Y. Edgeworth on 



To appreciate the order of error which may arise from these 

 inaccuracies, we may proceed as in my paper of last Octo- 

 ber*. First, let us confine our attention to the Mean, sup- 

 posing for a moment the Modulus accurate. Let k have 

 been determined according to Prof. Chauvenetfs method, so 



that 



6 



<sh 



2m-l 

 2m ' 



To determine more accurately the probability of an observa- 

 tion not exceeding a we must put for a, a-\-z, where z is the 

 error of the Mean subject to the law of frequency 



s/ 



m 



y- 



~e 



The proper course is therefore to evaluate the expression 



•/-» X C ' V 7TC 



Expanding 6, and neglecting the higher powers of z\, we 



have for the correction of #/- } the subtrahend —j=~ e~ fi2 i 



\c J Virn 



where /3 is put for -. Call this modification of 6. ~dd. To see 

 c 



how the prima facie limit /3 is affected by this modification, 



let us put 2—1 





L v-rvvj\H-r^HJ— ^n ' 



whence 



B<9(/3)+A/3x0'(/3)=O. 



Whence 





an extension of the limit which may be sensible when n is 

 small. 



In the example given by Prof. Chauvenet the uncorrected 

 limit as found by him is 1*22. This divided by the Modulus 

 [which= N/2e=*8] is 1'5. This result, our /3, divided by 15 

 the number of observations, gives "1 as the correction of fi ; 

 •08 as the correction of the limit a. The limit must be ad- 

 vanced to 1-30. This does not come up to the discordant 

 observation 1'40. But we have still to take into account 

 that we have been employing only the apparent Modulus (and 

 Mean Error), not the real one. In virtue of this consideration 

 I find — by an analysis analogous to that given in the paper 



* Phil. Mag. 1886, vol. xxii. p. 371. t See the paper referred to. 



