124 THEORY OF STATISTICS. 
geometric mean is always determinate and is rigidly defined. The 
computation is a little long, owing to the necessity of taking 
logarithms: it is hardly necessary to give an example, as the 
method is simply that of finding the arithmetic mean of the 
logarithms of X (instead of the values of X) in accordance with 
equation (11). If there are many observations, a table should be 
drawn up giving the frequency-distribution of log X, and the 
mean should be calculated as in Examples i. and ii. of § 9 and 10. 
The geometric mean has never come into general use as a repre- 
sentative average, partly, no doubt, on account of its rather 
troublesome computation, but principally on account of its some- 
what abstract mathematical character (cf. § 4 (c)): the geometric 
mean does not possess any simple and obvious properties which 
render its general nature readily comprehensible. 
23. At the same time, as the following examples show, the 
mean possesses some important properties, and is readily treated 
algebraically in certain cases. 
(a) If the series of observations X consist of » component 
series, there being IV, observations in the first, &V, in the second, 
and so on, the geometric mean G of the whole series can be 
readily expressed in terms of the geometric means @;, G@,, etc., of 
the component series. For evidently we have at once (as in § 13 
(®))— 
N.logG=0N,log G+ Ny, logG+ .... +N, log@,. . (12) 
(6) The geometric mean of the ratios of corresponding observa- 
tions in two series is equal to the ratio of their geometric means. 
For if 
X= 1/ Xo, 
log X =log X; —log X,, 
then summing for all pairs of X;’s and X's, 
G=0G,/G, : --9(13) 
(c) Similarly, if a variable X is given as the product of any 
number of others, z.e. if 
X= Xa vid. 20, 
xX, X, .... JX, denoting corresponding observations in 7 
different series, the geometric mean G' of X is expressed in terms 
of the geometric means @;, Gi. =. 0. Gof X,, X,, . ..*. X,, by 
the relation 
= 0, CN 0, . (14) 
That is to say, the geometric mean of the product is the product 
of the geometric means. 
sg