An Introduction to the theory of statistics

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Monograph

Identifikator:: 1008917265

URN:: urn:nbn:de:zbw-retromon-19129

Document type:: Monograph

Author:: Flückiger, Otto http://d-nb.info/gnd/117736708

Title:: Die Schweiz

Edition:: Zweite Auflage

Place of publication:: Zürich

Publisher:: Druck und Verlag von Schultheß & Co.

Year of publication:: 1914

Scope:: 1 Online-Ressource (VI, 243 Seiten)

Digitisation:: 2017

Collection:: Economics Books

Usage license:: Get license information via the feedback formular.

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Document type:: Monograph

Structure type:: Chapter

Title:: Bevölkerung

Collection:: Economics Books

XVIL.—NORMAL CORRELATION. | array of z, or of z, in its mean, and as the distribution of every array is symmetrical about its mean, RR must bisect every horizontal chord and CC every vertical chord, as illustrated by the two chords shown by dotted lines: it also follows that RR cuts all the ellipses in the points of contact of the horizontal tangents to the ellipses, and CC in the points of contact of the vertical tangents. The surface or solid itself, somewhat truncated, is shown in fig. 29, p. 166. 6. Since, as we see from fig. 50, a normal surface for two correlated variables may be regarded merely as a certain surface for which » is zero turned round through some angle, and since for every angle through which it is turned the distributions of all x, arrays and x, arrays are normal, it follows that every section of a normal surface by a vertical plane is a normal curve, ze. the distributions of arrays taken at any angle across the surface are normal. It also follows that, since the total distributions of x and x, must be normal for every angle though which the surface is turned, the distributions of totals given by slices or arrays taken at any angle across a normal surface must be normal distributions. Rut these would give the distributions of functions like a.z,+b.x, and consequently (1) the distribution of any linear function of two normally distributed variables x; and z, must also be normal ; (2) the correlation between any two linear functions of two normally distributed variables must be normal correlation. To find the angle § through which the surface has been turned, from the position for which the correlation is zero to the position for which the coefficient has some assigned value r, we must use a little trigonometry. The major and minor axes of the ellipses are sometimes termed the principal axes. If &, & be the co- ordinates referred to the principal axes (the &-axis being the x, axis in its new position) we have for the relation between £5 &y xy, x, the angle 6 being taken as positive for a rotation of the z-axis which will make it, if continued through 90° coincide in direction and sense with the z-axis, § =x,. cos 0+x,. sin 0 8) §y=a,. cos 0 — z,. sin 6} ( But, since ¢; &, are uncorrelated, 2(£,¢6,)=0. Hence, multiplying together equations (8) and summing, 0= (03 - 03) sin 26 + 25.00, cos 20 27.000, tan 26 = He 321 (9) 01

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