ON VARIATIONS IN THE VALUE OF THE MONETARY STAMDARD. 199 



fied in the manner above described, and a new (erroneous) index-number 

 is deduced. 



In this table the first column contains the names of articles in the 

 order adopted by Mr. Palgrave in his table 27. The second column 

 contains the ' weights ' assigned by him under the heading of ' Relative 

 Importance.' The third column consists of multipliei's formed by adding 

 twenty digits at raudom, and thus calculated to deflect the weights from 

 their respective true values to the extent of, say, 12 per cent, on an 

 average. The fourth column gives the new system of weights thus 

 affected with error. The fifth column contains (comparative) prices 

 taken from Mr. Palgrave's table 26. The sixth column furnishes a new 

 set of multipliers assigned by chance. The seventh column gives the 

 prices affected by error, and multiplied by 90 (the average value of 

 the chance-multipliers). The eighth column gives the product of the 

 ei'roneous weights and the erroneous prices ( x 90). The sum of this 

 last column, 1,413,480,000, divided by ninety times the sum of the 

 erroneous weights, which sum is 172,486, gives the erroneous index- 

 number 81 ; whereas the true index-number, on the assumption here 

 made that Mr. Palgrave's data are absolutely correct, is, as computed by 

 him, 76.^ 



Thus the falsified result is too great by -j^, or about 6 or 7 per cent. 

 That is a result quite consonant with the theory wliich assigns such a 

 measure of the error to be expected ^ that the result is as likely as not to 

 be out by 4 per cent., and that the odds are only five to one against the 

 error being so large as 8 or 9 per cent. It would have been nothing 

 miraculous if the result had been out by sixteen per cent. ; nothing more 

 extraordinary than, for instance, the fortuitous sequence which may be 

 observed in our third column of eight random aggregates falling below 

 the average about which they should oscillate, namely 90.^ 



The same table furnishes another verification, if, making abstrac- 

 tion of Mr. Palgrave's weights, we assume the index-number calculated 

 on the principle of the economist to be correct, and regard the figures in 

 our sixth column as erroneous weights (the true weights being all equal). 

 Upon this understanding we have the true result, the Simple Arith- 

 metic Mean of the comparative prices, 75"1 ; whereas the erroneously 

 Weighted Mean is 75'7, that is, it is in excess by about '8 per cent. Now 

 the measure of error here predicted by theory* is such that an error of 

 ■7 per cent, is as likely as not to occur. The occurrence of "8 per cent, 

 is therefore eminently consonant with the theory.''' 



' Third EepoH on Bepression of Trade, Appendix B. Memorandum by R. I. Pal- 

 grave. Tables 26 and 27. 



^ Taking 8-5 as the Modulus of the resultant error. See above, p. ]97. 



^ The probability of an error exceeding 1-9 times its modulus is -0072. The prob- 

 ability of the sequence referred to is -0078 ( = 57). 



1 /sv - SE - 



* By case (1) above, p. 11, the modulus is -7- x a / ' - x k. Here n is 19 ; - — ' 



■^''^ V n n 



is found to be -08, and k is -21. Whence the modulus is about -Ol-l, or I'o per cent. 



' Perhaps it may be asked here whether the example given is suited to exemplify 

 our estimate of the third species of error (see above, p. 193) : that due to the total 

 omission of certain articles. The answer is that this estimate, involving a larger 

 element of induction, does not profess to be so amenable to veriti cation as those 

 which are derived from known and steady ' sources of error,' like our aggregate.-^ of 



