ON VARIATIONS IN THE VALUE OF THE MONETARY STANDARD. 291 



But thougli in the present operation the weights would be much the 

 same as before, the balance, the method of combination, is different. In 

 view of the evidence adduced in the last section that price-variations 

 are apt to be grouped asymmetrically, the ' arithmetic ' species of mean 

 becomes precarious when our qucesitnm is a quasi- objective type. The 

 additional complexities which have been introduced iu this section make 

 against the geometric mean which was above recommended a certain 

 hypothesis. There exists another species of mean more adapted to the 

 rough character of our calculation, the Median ; that is, in the simpler 

 cases, that quantity which has as many of the given observations above 

 it as below it, but a certain analogue of this operation, when the obser- 

 vations have different weights. The required formula is the Weighted 

 Median, the operation designated by Laplace,' as the ' Method of 

 Situation.' 



The reasons in favour of the Median may be thus summed up. If, in spite of 

 the evidence above adduced, the normal probability-curve should after all turn 

 out to be the most appropriate representative of the group under treatment, the 

 Median is a reduction well adapted to this case, affected as it is with a probable 

 error only slightly larger than the arithmetic mean (Laplace, loc. cit. See ' Pro- 

 blems in Probabilities,' Phil. Mag. Oct. 1886). But if the gi'oupiug is of the 

 geometrical (Galton-Macalister) species, the Median is still a very good reduction, 

 coinciding as it does with the greatest ordinate of the curve denoted. Moreover, it 

 has been shown by the writer (' On the Choice of Means,' Phil. Mag. Sept. 1887) 

 that there is a peculiar propriety in the use of the Median when the observations 

 are ' discordant,' when their facility-curve may be regarded as a compound made 

 up of dirt'ereot families, or different members of the same family, of symmetrical 

 curves. It is now to he added that this prerogative of the Median is retained 

 when some or all the discordant elements are of the geometrical species. Now 

 the phenomenon of ' discordance ' is remarkably evidenced by the different degTees 

 of dispersion which series of (c.(/., yearly) price-returns present in the case of 

 different commodities. Cotton, for instance, appears to have a much larger modulus 

 of fluctuation than Pepper. Add that this method of reducing observations is the 

 least laborious of all, and there will remain no doubt that in the present state of 

 our knowledge, and for the purpose in hand, the Median is the proper formula. 



The method of the "Weighted or Coi-rected Median may best be described 

 by an example. The first column of figures given below are price-varia- 

 tions, expressed as percentages, for nineteen commodities, obtained by com- 

 parison of the year 1870 with the period 1865-9. The figures are taken 

 from table 26 of the Appendix to the Memorandum contributed by Mr. 

 Palgrave to the Third Report on the Depression of Trade. The per- 

 centages given by him are here rearranged in the order of magnitude. 

 Opposite each percentage in the third column is given the proportional 

 quantity of commodity, or ' relative importance,' taken from Mr. Pal- 

 grave's table 27 (year 1870). The fourth column contains the (approxi- 

 mate) square roots of these quantities.-^ Now for the simple Median the 



not appreciably affect the result ; as may be seen by comparing the results correspond- 

 ing to two different systems of weights (see note 2 on this page). 



' Thcorie Analytique, Supplement 2. See the jjresent writer's paper on 

 Observatlnns relating to Several Quantities in 'Hermathena' (Dublin), 1887. 



- The quantities of commodities taken as weights correspond to the squares of 

 Laplace, ^y,, y^„, j)^, &c {loo. cit.) If we determine the Median by waj- of the third, 

 instead of the fourth, column, we in effect assign for our system of weights the 

 squares of the masses. This operation, indicated by the bars in the thu-d column, 

 gives 91 as the Median. It is interesting to observe how small is the difference 

 produced by the change of system — small in relation to the error incident to any 



U2 



