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

THEORY OF STATISTICS. difference to some quantity of the same dimensions, e.g. the semi- interquartile range. Our measure would then be, taking the skewness to be positive if the longer tail of the distribution runs in the direction of high values of JX, skownoss = (Gam 20 = (hi ~@)_Q+Q,-208 Q Q This would not be a bad measure if we were using the quartile deviation as a measure of dispersion : its lowest value is zero, when the distribution is symmetrical ; and while its highest possible value is 2, it would rarely in practice attain higher numerical values than +1. A similar measure might be based on the mean deviations in excess and in defect of the mean. There is, however, only one generally recognised measure of skewness, and that is Pearson’s measure (ref. 9)— mean — mode Hick standard deviation 4 (17) This is evidently zero for a symmetrical distribution, in which mode and mean coincide. No upper limit to the ratio is apparent from the formula, but, as a fact, the value does not exceed unity for frequency-distributions resembling generally the ideal distributions of fig. 9. As the mode is a difficult form of average to determine by elementary methods, it may be noted that the numerator of the above fraction may, in the case of frequency-distributions of the forms referred to, be replaced approximately by 3(mean — median), (¢f. Chap. VIL §20). The measure (12) is much more sensitive than (11) for moderate degrees of asymmetry. 27. The Method of Percentiles.—We may conclude this chapter by describing briefly a method that has been largely used in the past in lieu of the methods dealt with in Chapters VI. and VII, and the preceding paragraphs of this chapter, for summarising such statistics as we have been considering. If the values of the variable (variates, as they are sometimes termed) be ranged in order of magnitude, and a value P of the variable be determined such that a percentage p of the total frequency lies below it and 100 - p above, then P is termed a percentile. If a series of per- centiles be determined for short intervals, e.g. 5 per cent. or 10 per cent., they suffice by themselves to show the general form of the distribution. This is Sir Francis Galton’s method of percentiles. The deciles, or values of the variable which divide the total frequency into ten equal parts, form a natural and convenient series of percentiles to use. The fifth decile, or value of the variable which has 50 per cent. of the observed values 150

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