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

132 THEORY OF STATISTICS. That is, if M, M,, M, be the respective means, M=0 5, (8) Evidently the form of this result is again quite general, so that if X=X +X + v hell ails +X, M=M, +t HY, +L... =H or {9 As a useful illustration of equation (8), consider the case of measurements of any kind that are subject (as indeed all measures must be) to greater or less errors. The actual measure- ment X in any such case is the algebraic sum of the true measurement X; and an error X,. The mean of the actual measurements M is therefore the sum of the true mean M, and the arithmetic mean of the errors M, If, and only if, the latter be zero, will the observed mean be identical with the true mean. Errors of grouping (§ 11) are a case in point. 14. The median.—The median may be defined as the middle- most or central value of the variable when the values are ranged in order of magnitude, or as the value such that greater and smaller values occur with equal frequency. In the case of a frequency-curve, the median may be defined as that value of the variable the vertical through which divides the area of the curve into two equal parts, as the vertical through M7 in fig. 22. The median, like the mean, fulfils the conditions (4) and (c) of § 4, seeing that it is based on all the observations made, and that it possesses the simple property of being the central or middlemost value, so that its nature is obvious. But the defini- tion does not necessarily lead in all cases to a determinate value. If there be an odd number of different values of X observed, say 2n+1, the (n+ 1)th in order of magnitude is the only value fulfilling the definition. But if there be an even number, say 2n different values, any value between the mth and (n+ 1)th fulfils the conditions. In such a case it appears to be usual to take the mean of the nth and (n+ 1)th values as the median, but this is a convention supplementary to the definition. It should also be noted that in the case of a discontinuous variable the second form of the definition in general breaks down: if we range the values in order there is always a middlemost value (provided the number of observations be odd), but there is not, as a rule, any value such that greater and less values occur with equal frequency. Thus in Table IIL, § 3 of Chap. VI, we see that 45 per cent. of the poppy capsules had 12 or fewer stigmatic rays, 55 per cent. had 13 or more ; similarly 61 per cent. had 13 or fewer rays, 39 per cent. had 14 or more. There is no number of rays "1 I)

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