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An Introduction to the theory of statistics

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Multivolume work

Identifikator:: 1895264332

Document type:: Multivolume work

Author:: Myers, Gustavus

Title:: Geschichte der großen amerikanischen Vermögen

Place of publication:: Berlin

Publisher:: Fischer

Year of publication:: 1916 -

Collection:: Economics Books

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Volume

Identifikator:: 1895266750

URN:: urn:nbn:de:zbw-retromon-242184

Document type:: Volume

Author:: Myers, Gustavus http://d-nb.info/gnd/10190651X

Title:: Geschichte der großen amerikanischen Vermögen

Volume count:: Bd. 1

Place of publication:: Berlin

Publisher:: Fischer

Year of publication:: 1916

Scope:: XL, 412 Seiten

Digitisation:: 2022

Collection:: Economics Books

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

IX.— CORRELATION. 177 Hence the sons of fathers of deviation « from the mean of all fathers have an average deviation of only 0-522 from the mean of all sons ; ve. they step back or “regress” towards the general mean, and 0-52 may be termed the “ratio of regression.” In general, however, the idea of a “stepping back” or “regression” towards a more or less stationary mean is quite inapplicable—obviously so where the variables are different in kind, as in Tables V. and VI.— and the term “ coeflicient of regression” should be regarded simply as a convenient name for the coefficients 4, and 6,, RR and CC are generally termed the “lines of regression,” and equations (6) the “regression equations.” The expressions “ characteristic lines,” ““ characteristic equations” (Yule, ref. 8) would perhaps be better. Where the actual means of arrays appear to be given, to a satis- factory degree of approximation, by straight lines, we may say that the regression is linear. It is not safe, however, to assume that such linearity extends beyond the limits of observation. 14. The two standard deviations 8,=0, n/1-12 8,=0, 1-12 are of considerable importance. It follows from (7) that s, is the standard deviation of (z-6,.7), and similarly s, is the standard deviation of (y — b,x). Hence we may regard s, and s, as the standard errors (root mean square errors) made in estimating « from y and y from « by the respective characteristic relations x=05.y y =bya. s, may also be regarded as a kind of average standard deviation of a row about RE, and s, as an average standard deviation of a column about CC. In an ideal case, where the regression is truly linear and the standard deviations of all parallel arrays are equal, a case to which the distribution of Table III. is a rough approximation, s, is the standard deviation of the z-array and s, the standard deviation of the y-array (cf. Chap. X. § 19 (3)). Hence s, and s, are sometimes termed the “standard deviations of arrays.” 15. Proceeding now to the arithmetical work, the only new expression that has to be calculated in order to determine 7 5,, 0, $x» and s, is the product sum 3(zy) or the mean product p. Asin the cases of means and standard deviations, the form of the arithmetic is slightly different according as the observations are few and ungrouped, or sufficient to justify the formation of a correlation-table. In the first case, as in Example i. below, the work is quite straightforward. Ezample i., Table VII.—The variables are (1) X—the estimated 12

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