438 On the Law of Error and Correlated Averages. 



dann zutreffen wiirde] when the quantities a v b v c x are proportional 

 to a 2 , b 2 , c 2 &c. ; which in general is not the case" *. 



Suppose that the two sets of measurements which we have 

 above adduced as examples had been made not on American 

 recruits and English adult males, but upon two organs a and 

 b, so related to a third c that c= ^/ a 2 + b 2 - The values of a 2 , 

 as we have shown, vary according to a law of error ; and so 

 do those of b" 2 . Accordingly, by universal admission the 

 sum a 2 + V 2 will vary according to the typical law ; and we 

 have shown that in general, if a quantity varies according to 

 this law, so also will its square root. Thus the Arithmetic 

 mean of the observed c's will fit the Arithmetic means of the 

 observed a's and 6's. 



The gist of the reasoning, it will be remarked, is that the 

 greater part of a group conforming to the law of error is apt 

 to be packed within limits which are narrow relatively to the 

 largest possible member of the group, and even the average 

 member ; in the symbols above used £ (the deviation) is small 

 relatively to X (the average) ; for the greater part of the 

 group, at least, up to some percentile near the extremities. 

 This is true by the Laplace-Poisson theory above adverted 

 to, even in the case most unfavourable for the argument 

 where X is measured from a point just below the least 

 possible ; as in the example given at p. 432. Even then it 

 would be safe to treat (X + £) 2 , or \ 7 X + f, as equal to the 



linear function X -f 2£, or >y/X + J . — f , respectively. But 



yX 



for natural groups the origin should perhaps be placed at 

 some distance below the smallest possible observation. The 

 smallest possible dwarf must be well above zero. Upon this 

 view the smallness of the modulus in comparison with the dis- 

 tance of the centre from the origin becomes more decided. 



As a matter of fact, the ratio of the modulus to the mean 

 value (the order of our f-f-X) is found upon an average of 

 several instances, taken from Mr. Galton's men* and Mr. 

 Weldon's shrimps t, to be from j 1 ^ to ^q. Mr. Galton in 

 some authoritative observations on this topic J assigns for the 

 ratio in question (in the case of human stature) j 1 ^. 

 [To be continued.] 



* Cf. Cournot, 'Tkeorie des Chances; Oh. X. Art. 123. Morselli follows 

 Cournot in attempting to demonstrate a priori the impossibility of 

 constructing a type involving numerous means of different organs (Metodo 

 in Anti-apologia, 1880, p. 26). 



f Proc. Koy. Soc. 1888 and 1892. 



| Phil. Mag. vol. xlix, p. 44. He gives for the ratio of the mean to 

 the probable error, u about -30." 



