VIL—AVERAGES. tJ 
it is simply calculated, its value is always determinate, its 
algebraic treatment is particularly easy, and in most cases it is 
rather less affected than the median by errors of sampling. The 
median is, it is true, somewhat more easily calculated from a given 
frequency-distribution than is the mean ; it is sometimes a useful 
makeshift, and in a certain class of cases it is more and not less 
stable than the mean ; but its use is undesirable in cases of discon- 
tinuous variation, its value may be indeterminate, and its algebraic 
treatment is difficult and often impossible. The mode, finally, 
is a form of average hardly suitable for elementary use, owing 
to the difficulty of its determination, but at the same time it 
represents an important value of the variable. The arithmetic 
mean should invariably be employed unless there is some very 
definite reason for the choice of another form of average, and the 
elementary student will do very well if he limits himself to its 
use. Objection is sometimes taken to the use of the mean in the 
case of asymmetrical frequency-distributions, on the ground that 
the mean is not the mode, and that its value is consequently 
misleading. But no one in the least degree familiar with the 
manifold forms taken by frequency-distributions would regard the 
two as in general identical ; and while the importance of the mode 
is a good reason for stating its value in addition to that of the 
mean, it cannot replace the latter. The objection, it may be noted, 
would apply with almost equal force to the median, for, as we have 
seen (§ 20), the difference between mode and median is usually 
about two-thirds of the difference between mode and mean. 
22. The Geometric Mean.—The geometric mean @ of a series of 
values X,, Xy, X;, . . . . X,, is defined by the relation 
EE NTE R . (10) 
The definition may also be expressed in terms of logarithms, 
log @= 1 3(log X) . (11) 
N 
that is to say, the logarithm of the geometric mean of a series of 
values is the arithmetic mean of their logarithms. 
The geometric mean of a given series of quantities is always 
less than their arithmetic mean ; the student will find a proof in 
most text-books of algebra, and in ref. 10. The magnitude of 
the difference depends largely on the amount of dispersion of the 
variable in proportion to the magnitude of the mean (cf. Chap. 
VIII, Question 8). Itis necessarily zero, it should be noticed, if 
even a single value of X is zero, and it may become imaginary if 
negative values occur. Excluding these cases, the value of the 
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