An Introduction to the theory of statistics

Bibliographic data

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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Chapter

Document type:: Monograph

Structure type:: Chapter

Title:: Part II. The theory of variables

Collection:: Economics Books

XIL—CORRELATION : MISCELLANEOUS THEOREMS, 211 Squaring both sides of the equation and summing, 3(2) = 3(ry?) + (2) £ 25(oy2,). That is, if » be the correlation between z; and xy, and o, oy, oy the respective standard-deviations, o?=0.2+ 0,2 + 2r.0y0, +: (1) If 2; and x, are uncorrelated, we have the important special case o?=02+ 0,2 : (2) The student should notice that in this case the standard- deviation of the sum of corresponding values of the two variables is the same as the standard-deviation of their difference. The same process will evidently give the standard-deviation of a linear function of an number of variables. For the sum of a series of variables XV, |. | | X, we must have ol=0 +0 : +0 + +024 20.000, + 275.000, BNE hE are 7, being the correlation beween X, and X,, r,, the correlation between X, and X,, and so on. 3. Influence of Errors of Observation on the Standard-deviation. —The results of § 2 may be applied to the theory of errors of observation. Let us suppose that, if any value of X be observed a large number of times, the arithmetic mean of the observations is approximately the true value, the arithmetic mean error being zero. Then, the arithmetic mean error being zero for all values of X, the error, say §, is uncorrelated with X. In this case if x, be an observed deviation from the arithmetic mean, « the true devia- tion, we have from the preceding 0,2=0.2+057 . . . (3) The effect of errors of observation is, consequently, to increase the standard-deviation above its true value. The student should notice that the assumption made does not imply the complete in- dependence of X and 8: he is quite at liberty to suppose that errors fluctuate more, for example, with large than with small values of X, as might very probably happen. In that case the contingency-coefficient between X and J would not be zero, although the correlation-coefficient might still vanish as supposed. 4. Influence of Grouping om the Standard-deviation.—The consequence of grouping observations to form the frequency distribution is to introduce errors that are, in effect, errors of

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