190 KKPOiix — 1888. 



mav be conceived as percentages obtained after the manner of Mr. Pal^rave (see 

 Table 26 of Memorandum in Appendix to ' Third Report of the Commission on 



Depression of Trade ') by multiplying the ratio qj-j-^-^ — dJ 100. Let us denote 



the opjjareMf price-variations, the erroneous observations, as ^,(1 + e,), js, (l + e„) 

 . p„(l + e„), where e,, e^ . . . e„ are each positive or negative errors, usually 



proper "fractions. Similarly let Wj, w.,, &c., be the real weights; and iv^^ (l + «i), 



.f^^ (1 + e.,^ &c., be the apparent, or erroneous, weights. 

 The index-number obtained from such data is 



tPj {l + fi)xpi (1 + e j + W.J (1 + fo) X po (1 + eg) + &c. ^ 



Wl (1 + f{) + W, (1 + €2) + &c- 



Alike in the numerator and denominator of this expression we may segregate 

 tlie correct and the errojieous portion ; and reason by the first of the principles 

 above mentioned that the incorrect portion is of a smaller order than the sum of 

 the correct terms (the number of observations being sufficiently great). Accord- 

 ingly it will be allowable to expand by Taylor's Theorem and neglect higher terms. 

 We shall thus obtain a simple expression for the error of the resultant index- 

 number in terms of the errors to which each class of the data is liable. 



This investigation may be broken up into three steps : we may consider suc- 

 cessively three cases in an order of increasing complexity. First (1) we shall 

 suppose that the weights only are liable to error. Then (2) we shall introduce the 

 circumstance that the observations, the price-variations, are themselves incorrect. 

 Lastly (3) we shall talce account of the fact that certain categories of articles may 

 be altogether unrepresented. 



(1) Under the first head we shall first consider the simple case when the weights 

 are really equal, though apparently somewhat unequal. In this preliminary case 

 the symbohc expression above written becomes simplified by the disappearance both 

 of the e's and the tda. Expanding and segregating the hetei'ogeneous elements in 

 the manner indicated, we may wi-ite our result thus : — 



P\ +Pl + ^'^^ J 1 . i^l^l +Pl^-i + ^^- _ f I + ^2 + ^^' \ 



n I Pi+p.2 + &c- n /' 



where the term outside the brackets is the correct index-number, and the difference 

 of the second and third terms within the bracket is the error of the index-number : 

 the relative error, as it may be called, or (if multiplied by 100) the percentage 



error in symbols — , if I is the correct index-number. The result obtained may 

 be written 



I « I ^sp nJ \ap w J 



In this expression call the factors of €„ f,, &c., respectively -Ej, -E2, &c. 



Then ^- the error whose magnitude we have to estimate, is - (Ejej -|- Ejej + &c.). 



I ^* 



To determine the probable and improbable limits of this quantity we require to 

 Ivnow the magnitude, or at least the average extent, both of the E's and the e's. The 

 former datuuT depends upon the dispersion of the observations (the price-variations) 

 .about their mean. For any E, e.g., 



fSp_ \ /Sp 



\sp nf isp hp 



n 



= the deviation or error incurred by the individual price-variation as compared with 



