VIL—AVERAGES, 129
Whence, 1/7 =02831, //=3'532. The arithmetic mean is 4'587,
or more than a unit greater,
If the prices of a commodity at different places or times are
stated in the form “so much for a unit of money,” and an average
price obtained by taking the arithmetic mean of the quantities
sold for a unit of money, the result is equivalent to the harmonic
mean of prices stated in the ordinary way. Thus retail prices of
eggs were quoted before the War as “so many to the shilling.”
Supposing we had 100 returns of retail prices of eggs, 50 returns
showing twelve eggs to the shilling, 30 fourteen to the shilling,
and 20 ten to the shilling; then the mean number per shilling
would be 12:2, equivalent to a price of 0-984d. per egg. But
if the prices had been quoted in the form usual for other commodities,
 we should have had 50 returns showing a price of 1d.
per egg, 30 showing a price of 0-857d., and 20 a price of 1-2d.:
arithmetic mean 0'997d., a slightly greater value than the harmonic
 mean of 0°984. The official returns of prices in India were,
until 1907, given in the form of “Sers (2:057 Ibs.) per rupee.”
The average annual price of a commodity was based on halfmonthly
 prices stated in this form, and “index-numbers” were
calculated from such annual averages. In the issues of Prices
and Wages in India” for 1908 and later years the prices have
been stated in terms of “rupees per maund (82286 1bs.).” The
change, it will be seen, amounts to a replacement of the harmonic
by the arithmetic mean price.
The harmonic mean of a series of quantities is always lower
than the geometric mean of the same quantities, and, & fortior,
lower than the arithmetic mean, the amount of difference depending
 largely on the magnitude of the dispersion relatively to the
magnitude of the mean. (Cf. Question 9, Chap. VIIL)
REFERENCES.
General.
(1) FEcENER, G. T. “Ueber den Ausgangswerth der kleinsten Abweichungssumme,
 dessen Bestimmung, Verwendung und Verallgemeinerung,”
 Abh. d. kgl. sdchsischen Gesellschaft d. Wissenschaften, vol.
xviii, (also numbered xi. of the 4Abk. d. math.-phys. Classe); Leipzig
(1878), p. 1. (The average defined as the origin from which the
dispersion, measured in one way or another, is a minimum : geometric
mean dealt with incidentally, pp. 13-16.)
(2) FECHNER, G. T., Kollektivmasslehre, herausgegeben von G. F. Lipps;
Engelmann, Leipzig, 1897. (Posthumously published: deals with
frequency-distributions, their forms, averages, and measures of dispersion
 in general : includes much of the matter of (1).)
(8) Z1zER, FRANZ, Die statistischen Mittelwerthe; Dunckerund Humblot, Leipzig,
1908 : English translation, Statistical Averages, translated with additional
 notes, etc., by W. M. Persons, Holt & Co., New York,1913. (Nonmathematical,
 but useful to the economic student for references cited.)

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