376 Mr. F. Y. Edgeworth's Problems m Probabilities. 



This error may be somewhat reduced, if we determine the 

 probable-error (for the given observations) not by the dis- 

 tance between the apparent Median and the apparent Quartile 

 point, but by the distance between the two Quartile points. 



l*9c 

 The Modulus for the error of c is in this case ^— — . 



Vn 



We have so far been investigating the error incident to 

 reasoning up to the character of the curve from a not very 

 large number of observations. Let us now consider the error 

 incident to the complete process of reasoning up from the 

 given observations and down again to a new observation. 

 First, let us suppose the Mean given, and calculate the error 

 incident to the assumption that Q ; , the apparent Quartile, is 

 the real one. Let it be required to determine the probability 

 that a subsequent observation will not exceed «Q'. Put 6{x) 



for —7= 1 e~ f2 dt. Then, if Qf -\-z be the real probable error, 



VttJo _ 

 the probability that an observation being in excess of the Mean 



will not exceed uOj is 6 ( .. r , n _ /^, ■ — r J . Now z occurs with 

 V2-097 (Q'+z)/ yz_ 



a frequency expressed by the facility- curve y= j — j=e * 2 , 



I'Wn 



nearly. We have therefore as the probability of the next 

 observation above the Mean falling within aQ', the expression 





dz. 



Q' 



Put *477 a = /3, and expand 6 in ascending powers of z. The 

 first term of the expanded expression is 



J_oo Virk 



the uncorrected value. The second term vanishes. The 

 third term is 



1 Z.2 /•» -i 2 _?! z 



= _^_^ (/3 3_^ )x | 2 



