Die deutsche Wirtschaft

The nature of capital and income

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Monograph

Identifikator:: 1689946490

URN:: urn:nbn:de:zbw-retromon-101093

Document type:: Monograph

Title:: Die deutsche Wirtschaft

Place of publication:: Berlin

Publisher:: Hobbing

Year of publication:: 1926

Scope:: 528 Seiten; Illustrationen

Digitisation:: 2020

Collection:: Economics Books

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Chapter

Document type:: Monograph

Structure type:: Chapter

Title:: 19. Die Gesundung der Währung in Deutschland

Collection:: Economics Books

APPENDIX TO CHAPTER XVI 407 A simple measure of the extent of variability displayed by such a series of deviations from the mean is what is called the “standard deviation.” This is a sort of average of the devia- tions—not the ordinary arithmetical mean, but the mean found by taking the arithmetical mean of the squares of the deviations and extracting the square root. The standard deviation which represents the above twelve individual devia- tions is thus, — + (= 1.924 (—.9) + (1)? CDE CDR (LDR (124 (2h (= 954 (Dy bie 1 (1) 12 which is .95. This “standard deviation” is used instead of other averages for several reasons. The arithmetical mean of the deviations about the mean is quite unavailable, because, unless it be reckoned by disregarding all minus signs (in which case the result is an illogical makeshift), it is zero; the standard deviation is very readily calculated, not by performing the operations indicated above, but by recourse to a theorem that the mean of the squares of the deviations about the mean is equal to the mean of the squares of the deviations about any other magnitude less the square of the difference between the mean and this other magnitude. The proof of this theorem is simple, and may be found in the books on probability. Applying it to the illustrated case, we first take the deviations, not about the mean, but about some other magnitude, say 5%. These deviations are 0,0,1,0,0, —1,0, 2,0, —2, —1,0. The squares of these are 0,0,1,0,0,1,0,4, 0, 4, 1, 0, of which the arithmetical mean is 4H, or .902. This is the mean of the squares of the deviations about the mag- nitude 5. From this we are to deduct the square of the difference between the mean 4.9 and the other magnitude 5, about which the deviations were measured. The difference is 1, its square is .01. Deducting this from .902 we have .892 as the mean of the squares of the deviations about the mean. The Square root of this is .95, which is therefore the standard deviation sought. Calculated by this method the standard deviation may usually be obtained in less than one tenth the time required by the direct method. TR RR A sk SSR

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