For the fourth purpose, the regulation of currency, the proper construction of the index number would seem to be as follows: (_a_) The "articles" of which the prices are taken into account should be both commodities and services; (_b_) both wholesale and retail prices should be used; (_c_) the relative importance of each article should be proportioned to the demand upon the currency which it makes. But here as in other parts of the subject theory halts a little, and statistics lag far behind theory. Considering the theoretical doubts and statistical difficulties which attend the determination of _weights_ proper to each purpose, there is much to be said in favour of assigning equal relative importance to all the items; especially if care is taken to include many articles such as _corn_, _cotton_, etc., which for any of the purposes which may be contemplated must be of first-rate importance. Such is the character of some of the principal index numbers which have been constructed--those of the _Economist_, of Jevons, of Soetbeer, and of Mr. Sauerbeck. In the construction of such an index number the use of the arithmetic mean is not imperative. Jevons employs the geometric mean. His reasons for preferring it are not very clear (the "Variation of Prices," _Currency and Finance_, p. 120).... The geometric mean has also the advantage of being less liable than the ordinary average to be unduly affected by extremely high prices (_Report of the British Association_, 1887, p. 283). The great objection to the geometric mean is its cumbrousness. There is another kind of mean which has some of the advantages of the geometric, and is free from its essential disadvantage; namely, the median ... which is formed by arranging the items in the order of magnitude, and taking as the mean that figure which has as many of the items above as below it. For instance the median of the forty-five percentages on which Mr. Sauerbeck's index number is based was, for 1892, 66; while the arithmetic mean was 68. It is difficult to see why the latter result is preferable to the former; if what is required is an _index_ of the change in general prices, not specially referred to any particular purpose, such as of securing a constant benefit to a legatee.