ON VARIATIONS IN THE VALUE OF THE MONETARY STANDARD. 275 



In this table the first column contains the percentage decrease for each article. 

 The next two columns contain the differences between the average decrease (27), 

 and the individual decreases. The modulus, or measure of fluctuation, is foimd 

 to be about 16. Hence, by a well-known theorem, the probable error of the 

 suni of n differences, n being large, tends to be ^n x 16 x -477 (a theorem 

 which does not assume that the diflerences are grouped according to a known 

 curve). Suppose, for instance, n = 9. The probable error of the sum of n differ- 

 ences taken at random should be about 23. This may be illustrated by actu- 

 ally taking some batches of nine, say the first nine, tallow to linseed oil, the last 

 nme, hnseed to hemp, and a central nine. The siun of the first set of differences 

 is - 51 + 25 = - 26. The sum of the second set of differences is - 47 + 55 = 8. The 

 sum of a third set, from Scotch pig to pimento, is- 58 + 29 = - 29 ; while if we put 

 out the Scotch pig and take in turmeric we obtain + 5. These observed results are very 

 consonant with the theory that the probable error is 2.3. Hence the probable error 

 of the 7)16(171 of nine differences is 2f. Meanwhile the probable error of any single 

 difference may be found by-observing that the 'quartiles,' in :Mr. Galtou's phrase, 

 occur on the one side between - 10 and -11, and on the other side between +6 

 and + 9, gi%ing a probable error of, say, 9. Or we may proceed more hypothetically, 

 and, assummg that the grouping (of the differences) "is conformable to the ' normal 

 type, find the probable error (-477 x modulus) about 8. Thus the displacement of 

 the single article is seen to exceed the mean displacement of several articles in 

 about the degree required by theory. 



We have taken the simple (arithmetical') mean. But much the same would 

 be true if we had taken - miy weighted mean of all prices, in particular the ideally 

 best, whose weights are ap^, bp^, &c. (provided at least those coefficients are not 

 extremely unequal). The de%-iation of the particular standard from the general 

 standard is apt to be so considerable that it does not much matter by what system 

 pf weights we determine the general standard. The unit best in the individual 

 interest is, as we have seen above (p. 274), l + e + €_^. The unit in the general 



interest is of the form 1 + e + ^^r^^^^t*^^' (putting A = ap^, and similarly B). 



The deviation of the former from the latter is of the form e. - ^^^ + ^^^ + ^^- 



A + B + &C. 

 Now, if e^, e^, &c., be on an average of the order e, then by the theory of errors their 

 weighted mean, the l atter part of the expression just ^^litten, wiU be of the 



order e - -_- ,^ *" ■ ' , an expression which tends to zero as the number of 

 (A + B + C + &C.) 



the coefficients is increased. The unavoidable discrepancy between the particular 



and general interest is therefore not likely to be much diminished by a more 



exact calculation of weights when those weights are numerous. — Q.E.D. 



Take, for example, the statistics above cited, where there are only sixteen 



items, and let us suppose the weights so disparate as the cardinal numbers 1, 2, 



... 16. If we based our unit on the simple arithmetic mean, we have e = -27, 



and for the Unit 1-27. Now this U7iit,a,s applied to each particular interest, is apt 



to be out by about -1, or 10 per cent. In the tallow interest, for instance, 1-17 



would have been the best unit ; if we legislated exclusively in the sugar interest 



the unit would be 1*36. Let us see now how these misfits would have been 



m ended by a more elaborate adjustment of the standard. The expression 



— \ — =^ ~ — ' becomes when A = 1, B = 2, &c., about -3. The correction then 



A + J5 + &c. 



upon the arithmetic mean -27 would be of the order -3 x -1 (e being of the order '1) ^ 

 that is, '03, or 3 per cent. This theorem may be verified by actually assigning 



' The probability-curve. 



^ See below, pp. 290, 291. 



' Assuming that each of the articles (tallow, sugar, &c.) is subject to the same 

 law of fluctuation, we may conclude (from an examination of the table) tliat the 

 average error for any article is 10 per cent. 



T 2 



