ON YARIATIONS IN THE VALUE OF THE MONETARY STANDARD. 211 



necessary to record the number of observations (by way of dots) in more 

 than 07ie of the extreme compartments. The Median is the hvenfy-third 

 figure in the order of magnitude, that is, 7'i. Proceeding similarly for the 

 year 1873, -we find the Median of Mr. Sauerbeck's forty-five price- 

 variations 109. 



Now let us try the efiect of weighting. Running my eye over some 

 pages of statistics, I assign the digits 1, 2, 3 as they occur to the price- 

 variations, which are in pell-mell order up to 70 ; between 70 and 80 in 

 the order of magnitude ; and above 80 not represented at all. The sum 

 of the whole second column thus formed is 86. The central point corre- 

 sponding to half that sum is at the foot of the fir.st half of the second 

 column, corresponding to the entry 72 in the first column. Accordingly 

 72 is the adjusted Median. I try another system of precision-factors 

 arbitrarily assigned. And still the Median is 72 ! 



The comparisons offered by Mr. Sauerbeck's materials are summed up 

 in the accompanying table : 



For estimating the extent of difference to be expected between two index- 

 numbers which overlap as to some of their items, the following formula is derivable 

 from the reasoning at p. 194. Let n be the number of items common to both 

 schemes, n' the number special to one, and n" to the other. Put x'' for the 



fluctuation of one system of weights, and x"~ fo^ the other ; and for 



S(e'-e")- 



(above, p. 204) put x'- Then for the modulus of the diflPerence between the com- 

 pared results we have 



^-^"^ V 2 w + n'V ^ 2 / n + n"V 2/ 

 where C, as before, is the measure of the dispersion incident to returns of com- 

 parative prices. In practice we may put for ^ , the ratio between the summed 



weights of the item special to each index-number and the sum total for all the 

 items. These fractions are derivable from the third and fourth columns in the last 

 table of the Appendix. In calcidating the fifth and sixth columns of tliat table 



p2 



