204 EEPORT— 1888. 



throughout the whole run all the largest weights had been attached to the articles 

 whose prices were contiuuallj' low, e.g., cotton, and (for the last few years at least) 

 silk and Jla.r, then the discrepancies (between the weighted and simple mean) 

 would have been rather larger than theory predicts. Thus, for the year 1885 I make 

 silk exchange weights with meat, and thus bring down the index-number to 64 ; a 

 discrepancy from the Arithmetic Mean which, if continued — as it probably would be 

 — from year to year, would be a little too great. Similarly, when wheat exchanges 

 weight with lecither, and cotton with indigo, the index-number works out to 92 — a 

 discrepancy of two moduli, which is much too large for a continuance. 



This sort of abnormality is less likely to occur in the case of our scheme, where 

 none of the weights are so large as some of Mr. Palgrave's. Still, before pressing the 

 theory, it is proper to examine whether the larger weights — in our case those of 

 meat, fish, and coal — are, from year to year, coupled with extreme price-variations. ^ 



Whenever law of this sort is discernible the doctrine of Chances hides its 

 inferior light, which is serviceable only in the night of total ignorance. The pure 

 theory of Probabilities must be taken cum grano when we are treating concrete 

 problems. The relation between the mathematical reasoning and the numerical 

 facts is very much the same as that which holds between the abstract theory of 

 Economics and the actual industrial world — a varying and undefiuable degree of 

 ■ consilience, exaggerated by pedants, igTiored by the vulgar, and used by the wise. 



II. Next let us compare our result with that obtained by using some 

 other system of weights, e.g., Mr. Sauerbeck's. In the annexed table, page 

 205, column 1 is the same as column 2 of the last table, containing Mr. 

 Sauerbeck's prices. Column 2 gives Mr. Sauerbeck's weights (for 1885) 

 reduced to percentages of the total weight assigned by him to the 

 twenty-one articles which ai"e common to him and the Committee. 

 For example, 61 is the weight actually assigned by him to wheat. This, 

 multiplied by 100, and divided by 559, the sum of all the weights assigned 

 by him to the twenty-one articles, gives 11 (nearly). 



It is an interesting result of theory that the difference to be expected between 



the two weighted inde.x-numbers (for the same twenty-one price-^-ariations) is 



. about the same as, or only a little less than, that between one of them and the 



Simple Unweighted Arithmetic Mean. This result is found by putting for — in the 



formula given above'- » where e', e", express deviations for the two systems 



of weights respectively. 



The comparison between the two systems is presented in the accom- 

 panying summary. The slightness of the difference between the com- 



' The effect of large weights combined with high prices is strikingly shown in an 

 index-number (attributed to Dr. Paasche) which is published in Conrad's Jahr- 

 Jiic/i^r, vol. xxiii. p. 171. There are twenty-two items, among which Rye obtains 

 about thirty per cent, of the total weight, and tlie Cereals generally (between whose 

 prices there is a certain solidarity) about seventy per cent. It is no wonder that in 

 the year 1868, when the price of the Cereals was exceptionally high, the Weighted 

 Mean should be 118, while the Simple Arithmetic Mean of the twenty-two compara- 

 tive prices is only 104. 



- The expression proves to be equal to '4. 



