§ 8.Reasons for preferring the Median for Practical Purposes

Practically, however, there is little if any advantage in 17 and 21 over 11 (or 12, which in the case of year-to-year comparisons amounts to the same thing) because (1) weighting is of little importance; (2) the more perfect weighting contained in formulæ 17 and 21 will seldom differ materially from that of 11 and 12, for any gain of precision would probably be less than the errors in measurement of the Q's, which are never exactly known; (3) the systems of 17 and 21 are practically far more laborious. In the end we must be guided largely by practical considerations except where the great labor and expense of computation may be disregarded. If in a practical spirit we examine the merits of the various formulæ, we shall, I believe, reject all formula except 9 and 11, and come to the conclusion that the best index number is the weighted median. It has no rival in ease of computation. The score of the median in the table (formula 9) is high, although it fails in test 2. Excepting this test it meets, partially or wholly, every other test. It therefore possesses some merit even on the theoretical side.

In passing, we may mention a feature of medians, although I am disposed to regard it as a fault. Edgeworth emphasized the fact that price dispersion upward always or usually exceeds the price dispersion downward. There is no limit to the former, but the latter is limited by zero. Statistical tests show clearly this asymmetry of dispersion.From this fact it has been argued that the best average should be one from which large deviations above it count no more than small deviations below it. This condition, whether good or ill, is not met by arithmetical averages, but is met by the geometric average and by the medianwhich, in fact, usually closely follows the geometric average. Edgeworth also argues that the median is superior when the variabilities of the various elements averaged are widely different.

Edgeworth concludes that "in the present state of our knowledge, and for the purposes on hand, the median is the proper formula."

As to methods of weighting, theoretical discussion with reference to test 2 shows that the weighting should be made on the basis of values sold in one or the other of the years compared.

It is easy to show that a system of weighting the median by given weights, that is, by counting each price ratio, not only once but a certain number of times (that number being the weight) will not affect the relative fulfillments of the tests as met by the simple median 9, which is the only median in the table. Edgeworth has shown that for all practical purposes a very rough system of weighting will suffice.Whether the weighting be according to the values p0Q0, etc., or p1Q1 etc., or p0Q1, etc., or p1Q0 etc., is usually of no practical importance whatever. If, then, we subordinate theoretical to practical considerations, the proper procedure would seem to be to select certain constants consisting of simple integers, and as near as may be to the values dealt with in the years considered. These weights need not be changed every year, but should be changed when the values (p1Q1) change very greatly.

If it be desired to have a quantity or trade-index number (Q1, or T1) as well as a price-index number (P1), we may likewise select as the form for Q1 the median. In other words, the indexes for p's and Q's are best selected independently of each other. It is true we there by abandon any absolute mutual consistency between the two, but we are now speaking of practical, not theoretical, considerations.

One of the great practical advantages of the median is its use in conjunction with "quartiles" or "deciles" to portray dispersion as well as averages. This method of showing dispersion about a mean is both easier to calculate, and capable of more detail, if detail be desired, than the method of Karl Pearson of the "Standard Deviation" about an arithmetical mean.

The final practical conclusion, therefore, is that the weighted median serves the purposes of a practical barometer of prices, and also of quantities as well as, if not better than, formulæ theoretically superior.

In spite, however, of the peculiar simplicity and ease of computation which characterizes the median, and in spite of Edgeworth's strong indorsement, it remains still almost totally unused, if not unknown. Wesley C. Mitchellhas used the median for price indexes more extensively than any one else. Professor Davis R. Dewey has used them for wages in his special Census report on that subject.