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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Chapter

Document type:: Monograph

Structure type:: Chapter

Title:: Part III. Theory of sampling

Collection:: Economics Books

oo 2 THEORY OF STATISTICS. It should be noticed that if we define the principal axes of any distribution for two variables as being a pair of axes at right angles for which the variables &, &, are uncorrelated, equation (9) gives the angle that they make with the axes of measurement whether the distribution be normal or no. 7. The two standard-deviations, say 2; and 2, about the principal axes are of some interest, for evidently from § 2 the major and minor axes of the contour-ellipses are proportional to these two standard-deviations. They may be most readily determined as follows. Squaring the two transformation equations (8), summing and adding, we have 212 =0+ 03 JX (10) Referring the surface to the axes of measurement, we have for the central ordinate by equation (7) , J y 12 = 2051 = 72) Referring it to the principal axes, by equation (3) Ta 120 ron But these two values of the central ordinate must be equal, therefore D= ay05(1 CE 5)’ (11) (10) and (11) are a pair of simultaneous equations from which 2, and Z, may be very simply obtained in any arithmetical case. Care must, however, be taken to give the correct signs to the square root in solving. 2; +2, is necessarily positive, and 2, — 2, also if 7 is positive, the major axes of the ellipses lying along &; : but if » be negative, 3; — 2, is also negative. It should be noted that, while we have deduced (11) from a simple consideration depending on the normality of the distribution, it is really of general application (like equation 10), and may be obtained at somewhat greater length from the equations for transforming co-ordinates. 8. As stated in Chap. XV. § 13, the frequency-distribution for any variable may be expected to be approximately normal if that variable may be regarded as the sum (or, within limits, some slightly more complex function) of a large number of other variables, provided that these elementary component variables are independent, or nearly so. Similarly, the correlation between two variables may be expected to be approximately normal if 3929

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