208 



REPOKT 1888. 



the sums of all the second columns. Thus, 16 + 12 + 8'5:=3do. Find the- 

 central figure of the total second column : that is the figure which as 

 nearly as may be has 18 for the sum of figures above it and below it. 

 This figure proves to be the 3 at the top of the second compartment 

 opposite 72. Then 72 is the required Mean. 



In the third column another system of precisions has been tried to 

 illustrate the effect of treating some price- variations as more typical of 

 the change in the value of money than others. Tossing up a coin, the 

 writer has stuck down (corresponding to each figure in the first column) 

 2 if heads turned up, 1 if tails. The sum of these arbitrary coefficients 

 of precision is 30, and accordingly the adjusted Median is the point inter- 

 mediate between 72 and the return next below in the order of magnitude^ 

 wliich proves to be 67. The adjusted Median is, therefore, 69'5. 



By operating similarly on the price-returns for 1873 (given above) it 

 is found that the Simple Median is 108, the Median adjusted by taking 

 account of quantities still lOS. 



The deviation between the Median and the Simple (or other) Arithmetic Mean 

 cannot, so far as the wi-iter knows, be formulated exactly. It diminishes with the 



number of observations, being of the order - .=^ A superior limit is given by the 



expression n/I + ^tt x Modulus of the observation; in our case say •!, or 10 per 

 cent.' This limit is probably very superior, as the following trials, in addition to 

 those given above, suggest : 



The thirty-nine figures are those above referred to, given by Jevons 

 at p. 4?4 of his Currency and Finance. The Geometric Means have been 

 cited again here in order to bring out the curious fact that the Median 

 seems to keep closer to the Geometric than the Arithmetic. This property 

 (which it would be desirable to verify more fully) is agreeable to the 

 theory, first advanced by the present writer so far as he is aware, that 

 prices are apt to group themselves in an unsymmetrical fashion after the 

 pattern of the annexed curve, whose ordinates indicate the frequency of 



each price-variation. In the year 1857, for instance, the smallest figure 

 was 91, the largest 247 ; while the Geometric, Median, and Arithmetic 

 Means were respectively 129, 127, and 134. There is some reason to 

 believe that the Geometric and Median — especially the latter — are more 

 apt to be coincident with the point at which the gi-eatest number of 

 returns cluster, the greatest ordinate of the curve. 



If then we take as our qu^situm that figure which would he presented 

 hy the greatest numher of price-variations in the complete series of returns 



' See the writer's paper in ' Problems in Probabilities,' Phil. Mag., Oct. 1886. 



