on VAKIATIONS IN THE VALUE OF THE MONETART STANDARD. 191 



the average of a whole set ; relative to (divided by) the average. Such a deviation 

 might be symbolised as —^, if we put^ for the average price-variation. 



We may now proceed in two ways : (a) we may either suppose the deviations 

 Ej, E^, ascertained for the particular year or epoch to which the calculation in 

 hand may refer ; (p) or we may seek a measure for general use, and available 

 without the trouble of examining the dispersion of the price-variations for a parti- 

 cular year. In either case we are to regard the e's as errors grouped in random 

 fashion about a mean, which is zero. The coefficient which measures the dispersion 

 of these errors, the modulus for the e-fluctuation, must be supposed knowable. 

 Call it K. 



(a) On the former understanding, we are to regard Ej, E^, &c., as known factors. 

 Accordingly by a well-known theorem we have for the modulus, which measures 

 the extent of the error, 



- (El €i + £3^2 + &c.), 



«\/^i 



^ + E„ + &C. X K, 



(/3) Otherwise we are to regard Ej, E,,, as samples, so to speak, taken from an 

 indefinite number — a complete series (in Dr. Venn's phrase) of E's. We must 

 suppose the coefficient of fluctuation, or modulus, for this series to be given by 

 prior experience. Let it be C. Then we may put as the most probable value for 



the measure or modulus of — , the error under consideration, 



1 C 



But this viost prohahle measure cannot safely be used as the best measure. We 

 must take into accoimt that the real measure may be larger, and accordingly that, 

 by adopting the measure described as ' most probable,' we may be underrating the 

 probability of each extent of deviation (from zero) to which the quantity 



- [EjEi + Ejfj, &c.] is liable. However, the error thus introduced is only of the 



order —/=, that is, the —r^th part of the magnitude to be evaluated. Now that 



vn vw 



degree of error has been already incurred by the neglect of the higher terms in 



the expansion of — . Accordingly it would be nugatory to apply correctives 



to the error now under consideration. 



We have now to introduce the circumstance that the weights, both real and 



apparent, difi'er from unity. It is easy to see that in the new expression for _ 



the coefficient of any weight-error e,. is '^rPs_^ ■ which may be put in the form 



Stop biv 



^— E'r, where E'^ is now the proportional deviation of p^ from the weighted mean 

 few 



of the p \ viz., -^. Accordingly the modulus of ^ becomes 



Sw 

 In evaluating the coefficient of < there are, as before, two courses. Either (a) 

 we operate upon the known values of E',, E'j, &c., for the particular year or epoch 

 with which we are concerned ; or (/3) we may make a general estimate based upon 



