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Monograph

Identifikator:: 898897718

URN:: urn:nbn:de:zbw-retromon-20842

Document type:: Monograph

Author:: Ecker, Alexander

Title:: Taxämter oder private Schätzungen?

Place of publication:: Essen-Ruhr

Publisher:: Verlag von W.F. Schulte

Year of publication:: 1913

Scope:: 1 Online-Ressource (74 Seiten)

Digitisation:: 2017

Collection:: Economics Books

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

5 THEORY OF STATISTICS. measurement. Instead of assigning to any observation its true value X, we assign to it the value X, corresponding to the centre of the class-interval, thereby making an error 3, where xX; = X Zs 0. To deduce from this equation a formula showing the nature of the influence of grouping on the standard-deviation we must know the correlation between the error 6 and X or X;. If the original distribution were a histogram, X; and & would be uncorrelated, the mean value of 8 being zero for every value of Xj : further, the square of the standard-deviatioh of 6 would be ¢2/12, where c is the class-interval (Chap. VIII § 12, eqn. (10)). Hence, if 0 be the standard-deviation of the grouped values X; and o the standard- deviation of the true values JX, Bam 0 go =g< 15s But the true frequency distribution is rarely or never a histogram, and trial on any frequency distribution approximating to the symmetrical or slightly asymmetrical forms of fig. 5, p. 89, or fig. 9 (a), p. 92, shows that grouping tends to increase rather than reduce the standard-deviation. If we assume, as in § 3, that the correlation between 8 and X, instead of 6 and X, is appreciably zero and that the standard-deviation of § may be taken as ¢?/12, as before (the values of 8 being to a first approximation uniformly distributed over the class-interval when all the intervals are considered together), then we have Sg g 01 19 . (4) This is a formula of correction for grouping (Sheppard’s correc- tion, refs. 1 to 4) that is very frequently used, and that trial (ref. 1) shows to give very good results for a curve approximating closely to the form of fig. 5, p. 89. The strict proof of the formula lies outside the scope of an elementary work : it is based on two assumptions: (1) that the distribution of frequency is continuous, (2) that the frequency tapers off gradually to zero in both directions. The formula would not give accurate results in the case of such a distribution as that of fig. 9 (8), p. 92, or fig. 14, p. 97, neither is it applicable at all to the more divergent forms such as those of figs. 15, et seq. 5. If certain observations be repeated so that we have in every case two measures 2; and x, of the same deviation #, it is possible to obtain the true standard-deviation o, if the further assumption is legitimate that the errors 6, and §, are uncorrelated with each other. On this assumption 2192

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