An Introduction to the theory of statistics

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Monograph

Identifikator:: 1751730271

URN:: urn:nbn:de:zbw-retromon-127610

Document type:: Monograph

Author:: Yule, George Udny http://d-nb.info/gnd/12910504X

Title:: An Introduction to the theory of statistics

Edition:: 8. ed. rev

Place of publication:: London

Publisher:: Griffin

Year of publication:: 1927

Scope:: XV, 422 S; Ill., Diagr

Digitisation:: 2021

Collection:: Economics Books

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Document type:: Monograph

Structure type:: Chapter

Title:: Part II. The theory of variables

Collection:: Economics Books

VIIL.—MEASURES OF DISPERSION, ETC. 135 But the sum of the deviations from the mean is zero, therefore the second term vanishes, and accordingly 2=c2+d2. (4) Hence the root-mean-square deviation is least when deviations are measured from the mean, 7.e. the standard deviation is the least possible root-mean-square deviation. 3(&2), or 3(f.&) if we are dealing with a grouped distribution and f is the frequency of & is sometimes termed the second moment of the distribution about 4, just as 3(¢) or 3(f.§) is termed the first moment (¢f. Chap. VII. § 8): we shall not make use of the term in the present work. Generally, 3(f.£") is termed the nth moment. 4. If o and d are the two sides of a right-angled triangle, s is a—— Fic. 2.. the hypotenuse. If, then, #/H be the vertical through the mean of a frequency-distribution (fig. 25), and AS be set off equal to the standard deviation (on the same scale in which the variable X is plotted along the base), S4 will be the root-mean- square deviation from the point 4. This construction gives a concrete idea of the way in which the root-mean-square deviation depends on the origin from which deviations are measured. It will be seen that for small values of d the difference of s from o will be very minute, since 4 will lie very nearly on the circle drawn through A/ with centre .S and radius SJ/: slight errors in the mean due to approximations in calculation will not, there- fore, appreciably affect the value of the standard deviation. 5. If we have to deal with relatively few, say thirty or forty, ungrouped observations, the method of calculating the standard deviation is perfectly straightforward. It is illustrated by the figures given below for the estimated average earnings of

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