PURCHASING POWER (a) In the measurement of the purchasing power of money in general it is argued that the prices of all commodities, not subject to regulation or monopoly, are equally significant. Theoretically the problem is one of pure sampling, and the precision of the result in any year is directly proportional to the square root of the number of independent terms included and inversely proportional to the mean dispersion of these terms from their average in that year. In practice, the terms are not completely independent, for the prices of related commodities influence one another, and the precision is thus reduced. Also, the greater the interval from the base year, the greater tends to be the dispersion and the smaller the precision. In periods when prices are changing rapidly, as in the years 1914 to 1923, the dispersion is usually considerable and the measurement loses accuracy.
There are three types of averages applicable to price-relatives: their arithmetic mean as described above for the Statist index number; the geometric mean of the same numbers, that is the nth root of their product if there are n commodities; and the har monic mean, which is the reciprocal of the arithmetic mean of the reciprocals. Thus, for two commodities whose prices were 6d. and I od. in the base year, and 1 s. and 2S. id. in any other year, the price-relatives are 200 and 25o, their A.M. is 225, their G.M. is V (200X 250) = 223.6, their H.M. is I : (Tiny = 222.2. If the latter year is taken as i oo, the relatives in the original base year are 50 and 4o, the A.M. is 45, and 45:100=100:222.2. The H.M. of the "forward" relatives gives the same measurement as the A.M. of the "backward" relatives; or, if the comparison is between two places the H.M. of the relatives when the first is taken as base gives the same measurement as the A.M. when the second is so taken.
A general change in the relation of currency to its use tends to affect all prices in the same proportion, and "if other disturbing causes may be considered proportional to the ratio of change of price they produce in one or more commodities, then all the individ ual variations of prices will be correctly balanced off against each other in the Geometric Mean" (Jevons, Investigations in Currency and Finance, 1884, pp. 121-122 ; see also Bowley, Economic Jour nal, 1921, p. 202). The Geometric Mean is therefore considered appropriate to this problem. It has the advantage that it gives less importance to extreme measurements than does the Arithmetic Mean. It has the further advantage that the comparison by its use of any two years is independent of the choice of the base year (see below).