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. 119 involve the crude assumption that the frequency is uniformly distributed over the interval in which the median lies. 17. A comparison of the calculations for the mean and for the median respectively will show that on the score of brevity of calculation the median has a distinct advantage. When, however, the ease of algebraical treatment of the two forms of average is compared, the superiority lies wholly on the side of the mean. As was shown in § 13, when several series of observations are combined into a single series, the mean of the resultant distribution can be simply expressed in terms of the means of the components. The expression of the median of the resultant distribution in terms of the medians of the components is, however, not merely complex and difficult, but impossible: the value of the resultant median depends on the forms of the component distributions, and not on their medians alone. If two symmetrical distributions of the same form and with the same numbers of observations, but with different medians, be combined, the resultant median must evidently (from symmetry) coincide with the resultant mean, z.e. lie halfway between the means of the components. But if the two components be asymmetrical, or (whatever their form) if the degrees of dispersion or numbers of observations in the two series be different, the resultant median will not coincide with the resultant mean, nor with any other simply assignable value. It is impossible, therefore, to give any theorem for medians analogous to equations (5) and (6) for means. It is equally impossible to give any theorem analogous to equations (8) and (9) of § 13. The median of the sum or difference of pairs of corresponding observations in two series is not, in general, equal to the sum or difference of the medians of the two series ; the median value of a measurement subject to error is not necessarily identical with the true median, even if the median error be zero, ¢.e. if positive and negative errors be equally frequent. 18. These limitations render the applications of the median in any work in which theoretical considerations are necessary com- paratively circumscribed. On the other hand, the median may have an advantage over the mean for special reasons. (a) It is very readily calculated ; a factor to which, however, as already stated, too much weight ought not to be attached. (&) It is readily obtained, without the necessity of measuring all the objects to be observed, in any case in which they can be arranged by eye in order of magnitude. If, for instance, a number of men be ranked in order of stature, the stature of the middlemost is the median, and he alone need be measured. (On the other hand

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