ON YAKIATIONS IN THE VALUE OF THE MONETARY STANDARD. 283 



Upon either view tlie practical rules for extricating the mean are much 

 the same. They may be arranged under two headings, relating (1) to 

 the form in which the given observations are to be combined ; and (2) the 

 relative importance to be assigned to the different observations. 



(1) As to the first point the general rule is that, in the absence of 

 special presumptions to the contrary, an arithmetical mean (or linear 

 function) of the given measurements is the proper combination.' That 

 is to say, if the different measurements are r^.r^, &c., each purporting to 

 represent one and the same object — in our case the appreciation or depre- 

 ciation of money — the proper combination of these data is — 



it'ir[ + yu2r2 + &c. _ 

 tVi + W2 + &c. 



where the factors u\, W2, &c., are weights, such that if Wf is greater than 2V.2 

 then Ti contributes more to the result than r,. 



This general presumption in favour of the arithmetic mean may, how- 

 ever, be rebutted by specific evidence in favour of some other mean, and 

 it is here submitted that in the case of prices there does exist such specific 

 evidence in favour of the geometric mean. 



It appears that prices group themselves about sl mean, not according to a 

 symmetrical curve like that which corresponds to - the arithmetic mean, hut 

 according to an unsymmetrical curve like ^ that which corresponds to the geo- 

 metric mean. Before adducing the empirical proof of this proposition it may he 

 well to consider what a prt07'i grounds we might have for preferring the geometric 

 mean. There are ' those who consider that the mere accumulation of agreeing 

 experiences can seldom suffice, without some antecedent probability, to establish 

 an inductive conclusion. 



It has been shown by Mr. Galton and others that the geometric mean is 

 adapted to a particular species ^ of observations, which may be described as 

 estimates. For instance, the estimates which diflerent persons (or the same person 

 at different times) might make of a certain weight would be likely to err more in 

 excess tlian in defect of the true objective weight, and in such wise as to render 

 the geometric mean of such a series of estimates the proper method of reduction. 

 This law of prizing may well extend to prices. The fluctuating estimates which 

 from time to time a person might make of the ^ utility of an object, as measured 

 by the quantity of some other object, e.(/., money, might well fluctuate according 

 to the law which has affinities to the geometric mean. So far then as changes in 

 price might depend vipon fluctuations in demand,' there is something to be said 

 in favour of our proposition. 



Again, there exists a simple reason why prices are apt to deviate much more 

 in excess than in defect : * namely, that a price may rise to any amount, but cannot 

 sink below zero.^ 



' The ground of this presumption is partly that the arithmetic mean is one of the 

 simjHest methods of combination ; partly that it is specially adapted to a species 

 of observation which is very extensive in rermn nahird, which may be said to 

 be always tending to be realised, the exponential law of error, or probability-curve. 



- The probability-curve. 



^ The curve described by Dr. Macalister in his paper on The Law of the Geometric 

 Mean in the PMlosujMeal Transactions, 1879. 



* G. C. Lewis as quoted by Dr. Bain in his Logic. 



^ Wherever the law of Fechner applies. See papers by Mr. Galton and Dr. 

 Macalister, Froc. Royal Soc. 1879. 

 " I.e., the ' final utility.' 

 ' Variations in what is technically called the demand-curve. 



* As in the annexed diagram. 



" That price should be, in Dr. Venn's phrase, a ' one-ended phenomenon ' may 

 raise a presumption in favour of an asymmetrical grouping, but by no means dis- 



