368 



Mr. D. McAlister on 



[Nov. 20, 



measurements will disagree among themselves, but on arranging them 

 in order of size they show a tendency to cluster round some medium 

 value. We are naturally inclined to infer that the true value of 

 the unknown, or typical member of the class, is not far from this 

 value. How to define and determine the appropriate medium in 

 various classes of measurement becomes thus a natural object of 

 inquiry. On examination we find that there is no strict and final 

 criterion applicable to all cases. We have to start with an empirical 

 assumption, more or less justifiable on general grounds, but not 

 capable of rigid proof. In the ordinary Theory of Errors, which deals 

 primarily with discrepant observations, the assumption made is re- 

 ducible to this : — that, on the whole, the best medium we can take is 

 the Arithmetic Mean of the discordant measures. This is equivalent 

 to the statement that errors (or differences from the truth) of equal 

 amount in excess or defect are equally likely to occur. In the class of 

 measures referred to by Mr. Galton, which are not of the nature of 

 instrumental observations, reason is given for thinking that, on the 

 whole, a better medium value is likely to be obtained by taking the 

 Geometric Mean of the discrepant measures. This is equivalent to the 

 assumption that measures are equally likely which bear to the truth 

 ratios of equal amount in excess or defect, so to speak. For example, 

 that measures which are respectively (y§§^)th and ( }°°° )th of the 

 truth are equally likely to occur. This paper seeks to develop some 

 consequences of this fundamental principle. 



The practical outcome is a method for the treatment of a series of 

 measures which naturally group themselves round their Geometric 

 Mean. This method may be briefly presented as follows : — 



Let the measures be 



a-'i) a? 2 , $3 x n . 



Take the (hyperbolic) logarithm of each measure, thus forming a new 

 series, say 



Ui, 2/a> Vs • • • y* m > where y r =log e x r . 

 Form the Arithmetic Mean (A.M.) of the a?'s, so that 



TO(A.M.)=2a>. 



Form the Arithmetic Mean of the y's : this will be the logarithm of 

 the Geometric Mean (G-M.) of the »'s, so that 



nlog (G.M.) = 2?/. 

 If then -^EE4 (log (A.M.) — log (G.M.)), we have in h a " measure of 



precision" of the scries of x's which we call the "weight." 



The series of y's may then be treated as a series grouped round its 



