ON VARIATIONS IN THE VALUE OF THE MONETARY STANDARD. 193 



1 





whence, aa SEr^ = n — , it appears that the influence of - may be neglected, 



n being supposed large. 

 We may therefore write 



Modulus of ^ = ^^"-y ^K ; 

 or, employing the notation which we had lately occasion to introduce : 



Modulus of = —r- X » /l X ^ X ___ X K. 



I Vii V -2 V2 



This formula may be employed to utUise present as well as past experience. If 

 we treat ^ and -_ as respectively the mean square of deviation obtained from 



the set of weights and price-returns entering into the index-number which we are 

 computing, we shall thus have an approximate formula more convenient than the 

 complete expression for the Modulus. 



(2) We have now to introduce the circumstance that each p is liable to an 

 error pe. Each element of error of the form Ilf,. is now aggravated by an element 

 of the form Pe^- Accordingly the modulus of the total error will be Vn - -t PV-, 

 where k and e are the moduli for the independent partial errors respectively, n is 



the coefficient of k in the expression for the modulus of -^ in case (2) aixl P" is 



easily seen to be equal to "''"-^'' . 



There may now be required, as before, a general formula applicable without 

 any examination of the prices and weights on a particidar occasion ; or without 

 other data than the coefficients expressing the dispersion of the prices and weights 

 respectively. With this view, employing the notation already explained, and 

 rejecting terms which may be shown to be of an inferior order, we may put for 



^g^,theexpression(l+|)(l+_). 

 Hence for the modulus of — in the general case we have 



3sVi-?Vt''^(i*t)' 



(3) So far we have been estimating the errors due to the weights and prices 

 of the articles which enter into our index-number not being accurate. We have 

 now to take into account that not only are all those articles misrepresented, but also 

 that certam other articles may be wholly unrepresented. For it is unlikely that all 

 the classes of products which ought by rights to enter into an index-number can, 

 even constructively, put in an appearance. 



We have now to superinduce the error due to such omission upon the errors 

 already estimated. To effect this we proceed in the same way as when compoimd- 

 ing the errors proper to our first and second headings. That is, we shall separately 

 evaluate for the third species of error its modulus squared, or Jluctuation, as the 

 present writer has proposed to term this important coefficient. Then we shall add 

 the third fluctuation to the sum of the two preceding : that is, to the square of the 

 formula given at the end of the second heading. 



To find the fluctuation proper to the third heading, let us begin with the simple 

 case in which the weights are aU equal. As before, let Sp represent the sum of the 



1888. ' ^ f ^ 



