194 REPOET— 1888. 



observed (comparative) prices ; let n be tbeir number ; and for '-^ put simple p. 

 Let S'^ be tbe sum, and w' the number, of tbe it7wbserved prices. Then the error 

 incurred by putting j) for tbe Mean of all tbe prices, tbe relative error —, is 



\ n + n' n I ' n + 7i' 



Tbe most probable value of tbis expression is zero ; while its fluctuation is 

 found to be, in terms and by methods alreadj' explained, 



1 2nn' ^„ 

 n {n + n'y- 



Now superadd tbe circumstance that the weights are various, dispersed about 

 their mean according to the modulus _;^. The effect of tbis attribute is to multiply 



tbe fluctuation last written by 1 + ^. The resulting expression is to be added to 



the square of the formula given at the end of heading (2). The effect of this 

 addition is to insert a new term under the last i-adical in the formula for the 

 Modulus. This new term is 



2nn' p2 

 {n + n'f- 



This formula will require modification, if there is reason to believe that the 

 omitted articles have not the same average weight as those which are included ; 

 for instance, if, as is likely, the omissions are many in number, but inconsiderable 

 in weight. 



It will be noticed that in passing from (tbe dispersion of) the observed prices 

 and weights to what has not been observed there is an inductive hazard greater 

 than is involved by solutions of cases (1) and (2) in their more exact form, and 

 while we suppose (as in the examples which will be adduced below) that the errors 

 of weight and price emanate from regular and stable sources, so as to admit of safe 

 prediction. 



As in case (2), we may suppose the coefficients x *nd C based either on prior 

 experience or on the data appertaining to the particular calculation which is in 

 hand. 



It will be observed that these coefficients do not contribute equally to the re- 

 sultant error represented by our formula. C, expressing the dispersion of the prices, 

 is more efficacious than Xj appertaining to the weights. Similarly c, the measure of 

 tbe error incident to the prices, affects the error of the index-number more than k, 

 the corresponding modulus of the weights. 



It is proposed now to illustrate the formulae which have been given by working 

 a few examples. In these examples the statistical materials, tbe prices and 

 weights, are taken out of Mr. Palgrave's Memorandum, from tables 26 and 27 

 respectively. The unstatistical arbitrary assumptions which will be made are that 

 any price, and likewise any weight, is as likely as not to be out, in excess or defect, 

 of the true figure by 10 per cent., but very malikely to be out by 40 per cent., or, 

 more exactly, that the apparent values fluctuate about the real one in conformity 

 with a modulus which is 21 per cent. 



Of tbe immense variety of cases which might be constructed by combining in 

 different ways the attributes which define the preceding paragraphs, it will be 

 sufficient here to discuss the most important case (2) of both weights and prices 

 subject to error — divided into two species, according as (a) we wtUise aU tbe data 

 special to the calculation in hand, or (j3) content oui-selves with the most summary 

 estimate. 



Let us appty these tests to Mr. Palgrave's computation of a weighted mean for 

 the year 1885 {3Iemwcmdum in Appendix to Third Report on tJi» Depression 

 of Trade). First, according to method (a), the expression for the (proportionate) 

 error due to a particular element of the index-number, the weight and price of a 

 particular commodity, is 



