ON VABIATIONS IN THE VALUE OF THE MONETARY STANDARD. 201 



is as likely as not, then it may be presumed that a deviation of 20 

 per cent, is not likely, of 30 per cent, very nnlikely. Upon this hypothesis, 

 according to the general formulEe above investigated, the error, or fortuitous 

 deviation from the ideal, to which the Committee's index-number is liable 

 is as likely as not to be as large as 2 or 3 per cent., but is unlikely 

 to be 6 per cent., and very unlikely to be 10 per cent. Now let us 

 entertain the more unfavourable and almost certainly extravagant 

 hypothesis that each datum is as likely as not to be out by 25 per 

 cent., and may just possibly ei'r to the extent of cent, per cent, (an 

 error which, if possible in excess, is almost inconceivable in defect). Upon 

 this hypothesis our index-number is as likely as not to be out 5 per 

 cent, but is not likely to be out by 10, and very unlikely to be out by 

 15, per cent. 



The presumption that our calculation is not likely to be far out is 

 confirmed by comparing the results obtainable by our method with those 

 obtained by other operators upon different principles. If the corapared 

 figures differ little from each other it is presumable that they differ little 

 from the true, the ideally best, figure : that which would be obtained if 

 "the data were perfect. 



The index-numbers which challenge comparison with that proposed 

 by the Committee may be arranged under four categories, namely : 



I. Those which are formed by taking the Simj^le Arithmetical Mean of 

 the given price- variation ; the principle of the Economist's index-number, 

 or rather what would be the principle of that operation if the prices 

 operated on had not been selected with some reference to the quantity of 

 the corresponding commodities. 



II. what may be called the Weighted Arithmetical Mean, each price- 

 variation being affected with a factor proportioned to the quantity of the 

 corresponding commodity, the principle adopted by the Committee. 



III. The Geometric Mean, as employed by Jevons. 



IV. The Median, proposed by the present writer as appropriate to 

 certain purposes.' It is (in its simplest variety) formed by arranging 

 the given price-variation {e.g., 98, 80, 88, 87, 85) in the order of magni- 

 tude {e.g., 80, 85, 87, 88, 98) and taking as the Mean the middle figure 

 (in the above example the third figure, i.e., 87). 



Under each of these headings it is desirable to supplement actual 

 verification with a priori reasoning based on the principles laid down in 

 the earlier part of the Memorandum. 



We may begin with the case (A) in which the price- variations are 

 supposed the same for the compared index-numbers. Later on (B) we 

 shall take examples in which both the price-variations and the mode of 

 combining them are difierent. 



A. 



I. Let us take the prices which are to hand for 21 (out of the 27) 

 items of our index-number in Mr. Sauerbeck's well-known paper on the 

 prices of commodities.- Let us form the Simple Arithmetic Mean of these 

 prices for the year 1885, and compare it with the Mean obtained by 

 applying our system of weights to the same prices. The operation is 



' See sect. ix. of Jlemorandum ' On the Methods of measuring Variations in the 

 Value of the Monetary Standard,' Brit. Assoc. RejJort, 1887. 

 '^ Journal of the Statistical Society, 1886. 



