An Introduction to the theory of statistics

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Monograph

Identifikator:: 1751730271

URN:: urn:nbn:de:zbw-retromon-127610

Document type:: Monograph

Author:: Yule, George Udny http://d-nb.info/gnd/12910504X

Title:: An Introduction to the theory of statistics

Edition:: 8. ed. rev

Place of publication:: London

Publisher:: Griffin

Year of publication:: 1927

Scope:: XV, 422 S; Ill., Diagr

Digitisation:: 2021

Collection:: Economics Books

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Document type:: Monograph

Structure type:: Chapter

Title:: Part II. The theory of variables

Collection:: Economics Books

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 125%

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