ON VARIATIONS IN THE TALDE OF THE MONETARY STANDARD. 203 



inequality of the weights. The measure or modulus of the discrepancy is, in our 

 notation, 



where n is 21 ; C is presumed (by a sufficient, but certainly not very copious, induc- 

 tion) to be from '2 to "3 ; and x is found to be about "O.^ 



It follows that of the observed discrepancies, '6 and 5, one is, a priori, move 

 likely than not to occur, and the other not unlikely. A rapidly increasing improb- 

 ability attaches to the higher degrees of divergence. 



Of course it must be understood that this theorem in Probabilities, this state- 

 ment of what will occur in the long run, is based upon the supposition that the 

 weights are distributed impartially among the price-variations. But if through- 

 out the whole run the largest weight is attached to the largest, or smallest, 

 observation, the then fortuitous character of the phenomenon is impaired. In 

 fact the ' long run ' of which the theory may be expected to be true is a series of 

 heterogeneous index-numbers not of consecutive years. Something of this sort 

 is observable in the case of Mr. Palgrave's Weighted Mean compared with the cor- 

 responding Simple Arithmetic Mean. The enormous weights attached to the 

 continiialiy low-priced Cotton and the continually high-priced Meat seem to afi'ect 

 the Weighted Mean abnormally. To effect the comparison, we must not take the 

 averages given in Mr. Palgrave's table 26, but those which are obtained by 

 omitting from that table the three items Cotton Wool, Cotton Yarn, and Cotton 

 Cloth, which do not occur in the compared table 27. The annexed comparison 

 does not present the appearance of pure chance. The discrepancies are rather less 

 in magnitude than the theory I'equires. For the modidus, as deduced from Mr. 

 Palgrave's system of weights, proves to be about 8o per cent, of the Mean 80 or 90 :- 

 that is about 7, corresponding to a probable error of about 3'6. The set of dif- 

 ferences above registered seems to ranore a little within the limits so defined. 



The reason is, doubtless, that the impartial sprinkling of the prices among the 

 weights, presupposed by theory, is not fulfilled in fact. Had it happened that 



• See above, p. 196, where the present ^nriter records the Mean Square of Deviation 

 for the price-variation of nineteen different articles (given bj' the Uconomist) in 

 different years. The Mean Square of Deviation for the figures given by Mr. Sauerbeck 

 seems to be much the same. Again, the writer has, with much the same result, 

 ascertained (by the Galton-Quetelet method) the quartiles for a few groups of 

 English prices, like those given bj' Jevons. For example, in the case of the thirty- 

 nine figures of the prices for prime articles in lS()0-62 comparative with 1845-50 

 (^Currency and Finance, pp. 51, 52) the quartile (half the interval between the tenth 

 and the thirtieth) proves to be 11, corresponding to a modulus of about 22 per cent. 

 If, however, we take in all the 118 articles given on the same page the quartile is 17. 

 The groups of thirty-nine on Jevons' page 44, so far as they have been examined, give 

 much the same result as the thirty-nine on pages 51, 52. Jevons himself gives 2^ 

 as the ' probable error ' incident to the Mean of thirty-six price-variations ( Currency 

 and FiTiance, p. 157) — corresponding to a probable eiTor of 15, a modulus of 30 

 for the individual price-return. Doubtless the dispersion may be expected to be 

 greater the more distant the base. If precision could be expected, it would be proper 

 to express the coefficient as a percentage of the mean price-variations at each 

 epoch rather than of the initial price or basis. 



2 See end of last note. 



