Der Handelskrieg von England, Frankreich und Italien gegen Deutschland und Österreich-Ungarn

Amerikas internationale Kapitalwanderungen

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Monograph

Identifikator:: 1689579730

URN:: urn:nbn:de:zbw-retromon-103299

Document type:: Monograph

Author:: Freese, Heinrich http://d-nb.info/gnd/118535153

Title:: Nationale Bodenreform

Edition:: Zweite Auflage

Place of publication:: Berlin

Publisher:: Druck und Verlag von U. Weichert

Year of publication:: [1926]

Scope:: XVI, 472 Seiten

Digitisation:: 2020

Collection:: Economics Books

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

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Document type:: Monograph

Structure type:: Chapter

Title:: 12. Rückblick und Ausblick

Collection:: Economics Books

VIL.—AVERAGES. 115 (8) If a series of IV observations of a variable X consist of, say, two component series, the mean of the whole series can be readily expressed in terms of the means of the two components. For if we denote the values in the first series by X; and in the second series by X,, 3(X) = 3(X) + 3(Xy), that is, if there be NV, observations in the first series and %, in the second, and the means of the two series be M/;, J, respectively, NM=N.M +N, M, . (5) For example, we find from the data of Table VI., Chap. VI, Mean stature of the 346 men born in Ireland =67-78 in. gy 2 2 741 % y Wales=6662 in. Hence the mean stature of the 1087 men born in the two countries is given by the equation— 1087. M = (346 x 67-78) + (741 x 66-62). That is, #/=66'99 inches. It is evident that the form of the relation (5) is quite general : if there are » series of observations xX, X, .... KX, the mean M of the whole series is related to the means M;, M, ... . M, of the component series by the equation NM=N.M, +N. M+ .... +N. M, . (6) For the convenient checking of arithmetic, it is useful to note that, if the same arbitrary origin 4 for the deviations ¢ be taken in each case, we must have, denoting the component series by the subscripts 1, 2, . . . r as before, (fH) =3(fp&)+3(fpb)+ . .. - +3(ME) (7) The agreement of these totals accordingly checks the work. As an important corollary to the general relation (6), it may be noted that the approximate value for the mean obtained from any frequency distribution is the same whether we assume (1) that all the values in any class are identical with the mid-value of the class-interval, or (2) that the mean of the values in the class is identical with the mid-value of the class-interval. (¢) The mean of all the sums or differences of corresponding observations in two series (of equal numbers of observations) is equal to the sum or difference of the means of the two series. This follows almost at once. For if X=X, +X, 3X) =23(X)) + 3(X,) a

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