ON VARIATIONS IN THE VALUE OF THE MONETARY STANDARD. 197 



To find the dispersion of the w's we proceed similarly. The arithmetical 

 •mean is for every year 2200 -^ 19, or 116 nearly. The ' differences ' are to he 

 ■formed hy subtracting this figure from each of the entries in Mr. Palgrave's 

 column headed Relative Importance. The sum of the squares of the differences is 

 to be divided by 19 for the absolute mean square of difference as it may be 

 called. This result, divided by 116-, gives the mean square of error relatively to 

 the mean weight. The values thus extricated for the years 1873, 1880, and 1884 

 respectively are, in round numbers. 354,000, 35,100, 357,000 : each divided by 

 255,664 ( = 19 X 116'^) ; whereof the mean value is 1-38. 



Substituting in the general or summary formula for the modulus of -= the values 



ifor C* and x" just ascertained, and for C and k the assumed value -21, we have 



-^ X V2^ X n/-05 X -044 + 1-05 X -044 = jtsr x 1"54: x -22 (nearly) = -077 ; 



whereas the answer found by the more exact method was -086. This consihence 

 seems greater than might have been expected, considering the small number of the 

 ■ elements entering into.the computation — only nineteen — and the scantiness of the 

 induction by which we determined the coefficients C and x- 



If we employ the summary formula as a short method of utilismg the data 



special to the index-number of 1885, we shall find that — as based upon the flue- 



tuationofpricesforthisyear is -08; and^ the mean square of deviation for the 

 •ws is stUi 1-38. Hence, as the approximate expression for the modulus, we have 



^ xl-54x-2lA/n6 = -08. 



4-36 



Thus we reach much the same result by the shorter as by the more tedious route. 

 We shall presently — in the portion of this paper addressed to the general 

 reader — try an experiment calculated to verify our deductive reasoning — so far as 

 a theorem in the Calculus of Probabilities can be verified by a single experiment. 

 We shall aflect each of the elements in Mr. Palgrave's index-number for 1885, 

 each weight and price, with a figure taken at random from a series of figures 

 hovering about unity in conformity with a modulus equal to -21. Such a series 

 -the writer happens to have ready to hand : consisting of sums of twenty digits 

 taken at random from mathematical tables, where the mean value is 90 and the 

 absolute modulus 19. The relative modulus, therefore, the modulus for the series 

 -when we divide each aggregate by 90, is -21. Accordingly it will be sufficient 

 to multiply each weight both in the numerator and the denominator with one of 

 the sums (of twenty digits) taken at random, and similarly affect each price enter- 

 'ing into the numerator, while the denominator is multiplied by 90. 



To resume now, in popular language, this somewhat teclinical inquiry. 

 The subject under investigation is the error to which our computation of 

 index-numbers is liable — the error relative to, or per cent, of, the true 

 value which we seek. We want to know, for instance, whether it is as 

 Uikely as not that our calculation exceeds (or falls short of) the correct 

 result by 10 per cent, of that result ; whether it is very improbable that 

 tbe excess (or defect) should be as great as 25 per cent. 



The error thus conceived is found to depend in a definite manner upon 

 six distinct circumstances. The erroneousness of the result is greater, the 

 ;greater the inaccuracy of the data, viz., the weights, and the (comparative) 

 prices. The erroneousness of the result is also greater, the greater the 

 ■inequality of the weights, and the greater the inequality of the price- 

 returns. Lastly, the result is more accurate, the greater the number of 

 the data, and the smaller the number of omitted articles. 



These circumstances are not all equally operative. Other things being 

 the same, the inaccuracy of the price-returns affects the result more than 



