ON YABIATIONS IN THE VALUE OF THE MONETAET STANDAED. 147 



mation and addition of the columns in Mr. Giffen's Table III., B. One 

 such operation is required to construct the denominator of the Index- 

 numbers which express the ratio between the level of prices in the standard 

 year and each other year. That denominator is in general terms, as we 

 have seen, y„ v^ + yb 7rj + &c. ; or, if we take the y's from one year, say x, 

 and the tt's from another year, say ij, the denominator becomes 



To be compared with this denominator there are (?i — 1) numerators one 

 for each of the years except the standard one, each numerator of the form 



where s is a year compared with y in respect of the level of price. 



There are, in general, then, n such operations: (n — 1) for the 

 numerators, and one for the denominator. But in the particular case 

 where the types of price and quantity are taken from the same year, 

 where x = y, the denominator reduces to 



Qax Pax + ^ix Pbx + &C- = Valuo for year re, 



a given figure which requires no computation. Accordingly one opera- 

 tion — that of calculating the denominator — is spared. Again, if we take tlie 

 factors from the particular year, say z, which is being compared in respect 

 of the level of prices with the standard year, that is, if in the last paragraph 

 we put X = z, the numerator reduces to 



qa,Paz + qb=lh^ + &'C., 



forming the total value for the year c, which is a given figure. But 

 meanwhile, in employing a different scheme of factors for each numerator, 

 we have necessitated the use of (n — 1) different denominators, each of the 

 form 



qa.zPa.z + qbzPb.+&C. 



The valuation of each of these forms will require (n — 1) operations of 

 the kind described. 



In addition to this slight advantage in respect of ease there may also 

 be ascribed a peculiar elegance to the selected formulte. But they have 

 no claim to the highest degree of accuracy. That distinction belongs to 

 a more complicated system, which is now to be described. We have so 

 far taken for granted that each typical quantity y is furnished by some 

 q which is the actual quantity for a particular year. But it is more agree- 

 able to the Calculus of Probabilities to take som.e Mean of the given q's for 

 the type y. This principle has been recognised in the First Report of 

 the Committee, where it is proposed that for the purpose of comparing the 

 level of prices at two different epochs the factors employed should be the 

 mean of the respective quantities. In the variety of the problem with 

 which we are at present concerned we may suppose a whole series of 

 corresponding quantities, e.rj., for the article a g,,,, ?u2, &c., q^n- The mean 

 of all or any number of these quantities may be taken for our type. 

 Now, out of n quantities (2" — 1) distinct combinations may be formed. 

 Instead, therefore, of the n diSerent arrangements of factors which we at 

 first found we have now 2" — 1; which, being combined with the one 

 peculiar scheme employed in Mr. Gifien's Table IV., makes 2". 



Similar remarks apply to the types of price. "We have so far taken 

 actual prices for our types. But it may be better to imagine a sort of 



L 2 



