ON VARIATIONS IN THE VALUE OF THE MONETARY STANDARD. IS^' 



least with any reasonable probability, exceed. But it is doubtful whetber 

 such a limit admits of being fixed with precision. The erroneousness of 

 the conclusion could only be ascertained by inference from the inaccuracy 

 of the premises. But it is difficult to appreciate with mathematical pre- 

 cision the error to which our data are liable. We may, however, argue 

 that, if the erroneousness of the premises is approximately of a certain 

 amount, if the error of the data is of a certain order, then the error of the 

 conclusion will be of a certain other order. 



Of course, it will be understood that, in attempting to evaluate 

 mathematically the error of our result, we mean its deviation from the 

 real numerical value of that quantity which we have here taken as our 

 qticBsitum : namely, the total national expenditure on material products at 

 any given time comparative with an initial epoch (abstraction being 

 made of any change in the total quantity of products which may have 

 occurred between the epochs).' The philosophical error which may be 

 committed by taking this sort of index-number as our ideal is the subject 

 of another sort of analysis. 



The subject of our investigation being thus defined, we may show 

 that the erroneousness of the result is less than that of the data. There 

 are two lines of proof converging to the truth of this theory. First, we 

 may reason a priori by the Calculus of Probabilities that the index-number 

 is subject to a smaller percentage of error than the weights and price- 

 variations (given or referred to in columns 5 and 6 of the table). 

 Secondly, this deduction may be verified by actual trial. We may assign 

 a certain set of weights and price- variations as correct, and construct 

 several sets of variants diverging from the ' correct ' figures in haphazard 

 fashion. Then, operating with each set of variant data, we may calculate 

 several variant index-numbers. These, it will be found, diverge less— 

 that is by a smaller percentage— from the correct index-number than any 

 set of variant data from the corresponding correct datum. 



The second part of the evidence cannot be fully appreciated without 

 the prior reasoning. By itself it conveys only a moiety of the truth. 

 Those who are content with that fraction of knowledge are advised to 

 skip the small type and close reasoning of the immediately following 

 paragraphs and to pass on to the more easily read lessons of experience. 



The index-numher which is the result of our calculation ia subject to a less 

 error than the data which enter into it, for two reasons. First: The numerator 

 and denominator of the fraction which constitutes the iudex-munher form each an 

 aggregate of elements or parts, whereof each element is suhject to a presumably 

 independent error. Now, by a well-known principle of the Calculus of Probabilities, 

 the percentage error of such an aggregate is less than the percentage error incident 

 to each element (or at least to an element of average erroneousness). This prin- 

 ciple applies to the errors both of the weights and the observations (price-variations). 

 The next consideration apphes only to the former class of data. An error in any 

 7veight affects both the numerator and denominator in the same direction, whether 

 of excess or defect, and thus is to a certain extent self-corrected. 



This reasoning may be exhibited more fully by the aid of symbols. Let us put 

 the series p-^, p^, &c. . . ■ p„ for the real price-variations. These price-variations 



the student of Probabilities may, it is hoped, preponderate over the disadvantage 

 that it suggests to the general reader a more gross, blameworthy, and avoidable nus- 



I take than is contemplated here. 



I ' For a more exact definition of the queesitum, see in the First Report of the Com^ 



mittee the formula for the ' Principal Standard.' 



