ON VAKIATIONS IN THE VALUE OF THE MONETARY STANDARD. 155 



in tho other expression of the form, value of a in f. Now, it has been 

 shown by the present writer in a memorandum on the Accuracy of 

 Index-numbers, published in the Report of the British Association for 

 1888, that, in forming a mean of any given set of quantities, the difference 

 between the results obtained by adopting diQ'erent systems of weights is 

 apt to be inconsiderable. This proposition has been established both by 

 reasoning from the theory of probabilities and by pretty copious examples. 

 It is shown that the divergence between the two results tends to 

 diminish as the number of (supposed independent) items entering into 

 the average increases ; the probable deviation being proportioned to the 

 inverse square root of the number of items. This tendency to evanes- 

 cence is resisted by three circumstances : the inequality of the given 

 items which are to be averaged, the inequality of the weights which con- 

 stitute the set or system which is regarded as true, and the largeness of 

 the difference between each weight in that one system and the correspond- 

 ing weight in the other system. It can be shown that the last two 

 circumstances are equivalent to, or, rather, are contained under, one attri- 

 bute, namely, the inequality of the weights in either system.' 



These criteria are now to be applied to the case before us. In the first 

 place we have a very large number of elements to deal with — much larger- 

 than the number of enumerated articles which enter into Mr. Giffen's 

 Index-numbers. For Sir Rawson Rawson's Index-number includes the 

 unenumerated as well as the specified articles. Thei'e is, therefore, a 

 strong prima facie presumption that the divergence between the two 

 compared expressions will prove to be unimportant; even smaller than 

 in the case of the Index-numbers compared in the paper referred to, the 

 number of items being larger here than there. 



Then, as to the counter tendencies. There is no reason to apprehend 



any fatal inequality in the ratios of the form .;-— ; — - — ■. — ^. At least it 

 ^ ^ ^ Bulk of a m x 



would only be in cases of articles where the bulks were very small that 

 such an influence need be apprehended, the ratio in such a case tending 

 to infinity. It is easy to see, however, that this tendency would be cor- 

 rected by the ' weights ' ; that such an article would not be likely to have 

 much efl'ect on the whole expression. There seems no reason to appre- 

 hend any much more marked inequality in comparative bulks than in 

 comparative quantities, which, as we know from Mr. Giflen's tables, aro 

 not fatally unequal. 



There remains tho twofold condition that the weights of either 

 system should not inter se be very unequal. The most serious violation 

 of this condition seems to be coal in the case of exports. It appears from 

 Sir Rawson Rawson's statistics that the bulk of coal takes up an inor- 

 dinate proportion of the total bulk of all commodities. Accordingly 

 he has very properly excluded coal from his Index-number. It is interesting 

 to observe that, as shown in Tables I. and VII., the inclusion of coal does 

 not, as a matter of fact, distort the result so much as might have been 



' The measure of ' inequality ' is the square root of the sum of squares of all the 

 weights in a system -f- their sum. The divergence between the results is directly pro- 

 portionate to this expression. If the weights are perfectly equal the factor reduces 

 to vnity-i- Vn. But suppose one weight preponderates over its fellows to such an 

 extent as to constitute half of the total mass, the remainder of which we may 

 imagine split up among a number of small weights ; the resulting expression is no 

 longer of the order 1 -j- -^^n, but equals, at least, \. 



