with those that have been made, and can use the indexes in a candid and intelligent manner. That "amiable weakness to take upon faith plausible figures that fill a pressing want" would not then be so common.

Should weights be fixed or fluctuating? By changing them a more accurate measure of importance is undoubtedly acquired, but changes in an index must then be interpreted not only in terms of prices but also in terms of weights. Conceivably, some sort of an average of relative importance over a period could be used, but if so the variations would be lost sight of. When chain-indexes are used, weights can be varied without confusion, since price changes from year to year only are measured. Such figures do not accurately measure changes over a period. The question cannot be answered in a word, and we shall not attempt to settle it. There is much to be said for the stability resulting from the use of fixed weights, and in actual practice necessity frequently requires that one be satisfied with such.

(5) Average of Relatives Index Numbers versus Actual Prices Aggregated

In the section devoted to The Base the question of the desirability of actual instead of relative prices was raised, and some of the reasons indicated which have prompted a return to the former kind. This problem may now be considered a little more fully. Two major questions are involved: First, how to reduce commodities quoted in widely different units and in different quantities to a common denominator in order that they can be combined - for price level would not be reflected in the change of a single commodity; and second, what system of weights to use. The first question until recently seemed insuperable. As the Bureau of Labor Statistics puts it:

"it would be a statistical absurdity to make index numbers for the different years from the yearly averages of the actual money prices of a ton of coal, a yard of calico, a hundredweight of live hogs, 144 boxes of matches, a pound of raw rubber, a gallon of turpentine, 50 square feet of window glass, a dozen cans of salmon, a barrel of petroleum, a yard of trouserings, a mule, a pair of boots, a bushel of beans, a thousand feet of pine lumber, a crosscut saw, a barrel of cement, a two-bushel bag, a thousand bricks, a ton of steel rails, a dozen teacups and a dozen saucers, a spool of thread, a pine door, a pound of cotton, a dozen cans of tomatoes, a pair of door knobs, a hundredweight of barbed wire, a hammer, a quintal of codfish, a 'set' of bedroom furniture, a ton of brimstone, a dozen eggs, an apothecary's ounce of quinine, a barrel of salt, a dozen kitchen chairs, a pound of beef, a pair of cotton blankets, a nest of three oak-grained tubs, 100 pounds of onions, a carving set, a bushel of potatoes, a dozen pairs of socks, a three-quarter-inch auger, a barrel of herrings, a troy ounce of silver, a box of raisins, a ton of hay, a dozen undershirts, a quart of milk, a thousand shingles, a yard of broadcloth, a ton of cotton-seed meal, a gross of wood screws, and a pound of plug tobacco." 1

Even to reduce the various units with the prices quoted per length, dozen, cubical contents, area, weight, etc., to prices per pound, or some other single unit, will not suffice. Left in this manner an index

"greatly exaggerates the effects of price changes in the rare, costly, and relatively unimportant articles, like opium and silver, and correspondingly minimizes the importance of price changes in common, cheap, and important articles, like coal, petroleum, and pig iron. It avoids the inaccuracies of the average of relatives by committing much graver inaccuracies." 2

To remedy this defect, however, the device is now adopted by the United States Bureau of Labor Statistics in the case of wholesale prices of weighting the price per pound of commodi

1 Bulletin of the United States Bureau of Labor Statistics, Whole Number 181, Wholesale Prices, p. 245.

2 Ibid., p. 246.

"1

ties by the amount of physical product placed on the market in 1909. In this way a relative of weighted aggregate money prices is secured the last completed year being the base adopted instead of an average of relative prices. The theory upon which the number is computed is that "what is wanted in wholesale-price indexes as well as in retail-price indexes is a measure for changes in the cost of a given bill of goods.' This purpose seems to be the one in which most people are interested and the sum of actual prices appears best fitted to establish it. Mitchell, after summarizing the advantages of aggregates of actual prices, has the following to say: "Now the weighted aggregate of prices is the best measure of change in the money cost of goods; it is better in several ways than the simple arithmetic mean of relative prices, and in addition it has all the merits of the latter form of average." 2

"Aggregates of money prices weighted according to the importance of the several articles are as easy to understand as arithmetic means of relative prices. They are less laborious to compute than any other form of weighted series, for no relative prices are used; the original quotations are multiplied directly by the physical quantities used as weights, and the products added together. They are not tied to a single base period; but from them relative prices can quickly be made upon the chain system or any fixed base that is desired, and these relative prices themselves can be shifted about at will as readily as geometric means. Hence they are capable of giving direct comparisons between prices on any two dates in which an investigator happens to be interested. Hence, also, they can be compared with any index numbers covering the same years, on whatever base the latter are computed. Their meaning is perfectly definite which is not always true of medians. They can not be made to give apparently inconsistent results like

1 "Wholesale Prices," Bulletin of the United States Bureau of Labor Statistics, Whole Number 181, p. 246.

2 Ibid., Whole Number 173, p. 92.

arithmetic means. When published as sums of money, they can be added, subtracted, multiplied, divided, or averaged in any way that is convenient. When weighted on a sound system, they can not be unduly distorted by a very great advance in the price of a few articles, and yet, unlike medians, they allow every change in the price of every article to influence the result. In fact, they combine most of the merits and few of the defects characteristic of the various methods of averaging relative prices."

IV. CONCLUSION

1

The discussion has been carried far enough to establish the fact that index number making and using are far from simple things. The complexity of the problem seemed to make it necessary to develop the various points in this chapter in order to bring before the reader the theoretical and practical considerations surrounding the topic. In most respects little more has been done than to call attention to the more important phases of the subject and to leave the student to verify them by reference to such painstaking and comprehensive studies as those of Fisher, Mitchell, and others. Some of the more important practical applications of the subject are outlined in the following chapter. The aim here is not a critique, but rather an exposition of the principles upon which a critique must be based. If an interest in index number making and using has been aroused, the main purpose of what has been written here shall have been accomplished. After all, the main reliance must be placed in the scientific spirit and integrity of both maker and user. If these are lacking, the use of statistics is without a logical defense.

1" Wholesale Prices," Bulletin of the United States Bureau of Labor Statistics, Whole Number 173, p. 91.

REFERENCES

Bowley, A. L.-Elements of Statistics, pp. 111-118; 217-229. Coats, R. H.-Special Report on Wholesale Prices in Canada, 1890-1909, inclusive (Ottawa, Can., 1910). This report contains a useful summary of the principal index numbers now compiled in the United States and in foreign countries. Fisher, Irving - The Purchasing Power of Money, Ch. 10. Hooker, R. H. "The Course of Prices At Home and Abroad, 1890-1910," in The Journal of the Royal Statistical Society, Vol. LXXV, pp. 1-36 (December, 1911).

Mitchell, W. C.- Business Cycles, pp. 112-139, on "The Representative Character of Index Numbers."

"Index Numbers of Wholesale Prices in the United States and Foreign Countries," United States Department of Labor, Bulletin Bureau of Labor Statistics, Whole Number 173, July, 1915. Part I, "The Making and Using of Index Numbers," pp. 5-114; Part II, "Index Numbers of Wholesale Prices in the United States and Foreign Countries," pp. 115–324. (This publication in the field of index numbers is epoch-making. It includes a complete study of the technique of the construction and use, as well as a descriptive account of current and past index numbers in this and in foreign countries.) Meeker, Royal - "Some Features of the Statistical Work of the Bureau of Labor Statistics," in The Quarterly Publications of the American Statistical Association, March, 1915, pp. 431–441.