200 REPORT— 1888. 



It might be desirable to apply this sort of test on a large scale to the 

 computation recommended by the Committee, and to prove by specific 

 experience the conclusions ■which are dedncible from the Theory of Prob- 

 abilities concerning the accuracy of any index-number. 



These conclusions cannot be stated in their most exact form until 

 the price- returns, as well as the weights which enter into the computa- 

 tion to be tested, are assigned. But even at the present stage of our 

 procedure, and without reference to the price-returns of a particular 

 year, we may approximately estimate the accuracy of index-numbers of 

 the kind proposed by the Committee. For the purpose of a rough esti- 

 mate it is enough to know the weights (which are assigned in the Second 

 Report of the Committee) and to utilise past experience concerning the 

 course of prices in this country. A certain datum,' which had better be 

 determined precisely from the price-returns from the particular year 

 to which the index-number relates, may be apjji-oximately obtained by 

 induction from the experience of past years. 



Eliciting the required da,tum from the prices recorded by the Econo- 

 mist,"^ we may provisionally assert the following propositions concerning 

 the accuracy of index-numbers such as the Committee has proposed. 

 These, it will be recollected, involve twenty-seven English price-returns 

 and twenty-seven assigned weights.^ 



(1) In such an index-number, if the weights alone are supposed sub- 

 ject to error, then the average error of the result, its erroneousness as one 

 may say, is twenty times less than the error to which each weight is liable. 



(2) If the price-returns alone are liable to error, the erroneousness of 

 the result is ahout four-a^id-a-half times less than that of each datum. 



(3) In the general case, when both prices and weights are liable to error, 

 then, if that error be the same for both species of data, the error of the re- 

 sult is still about four-and-a-half times less than that same. If the error 

 of the weights become twice as great as that which is incident to the 

 prices, other things being the same, the error of the result is not materially 

 increased. The error of the weights would need to he five times as great as 

 that of the pi'ices in order to increase the error of the result by 50 per cent, 

 (making it only three times less than the error incident to the prices alone). 



The practical conclusion from these propositions ajipears to be : Take 

 more care about the prices than the weights. 



More detailed statements cannot be made without some assumption 

 as to the degree of inaccuracy to which our data are liable, the extent to 

 which our estimates of weights and prices deviate from the figures which 

 would be assigned if our knowledge and theory were perfect. In enter- 

 taining any suppositions as to the extent of this discrepancy, it is proper 

 to conceive that the lai'ger deviations, the more extensive errors, are less 

 frequent in the long run, or more improbable. Thus, if we suppose that 

 a deviation of each datum, weight or price, to the extent of 10 per cent. 



digits. Moreover, such verification as the theory admits would require a larger 

 number of items than the table in the text contains. For in general it must be 

 assumed that the numbers both of the included and excluded articles are large. Now 

 it is impossible to carve two sets of ' large numbers ' out of nineteen. 



' The coefiicient C defined above, -p. 191. 



- As given in Mr. Palo-rave's table 26 (see above, p. 196). 



s Namely, 5, 5, 5, 5 ; 10, 2i, 1\ ; 2\, 2|, 9, 2^, 1, 2i ; 2i, 2^, 2J, 2i ; 10, 5, 2i, 2^ ; 



3, 1, 1, 3, 1, 1. Whence the value of /c-;;;.. (see above, p. 193) is found to be 'Oo. 



