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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Chapter

Document type:: Monograph

Structure type:: Chapter

Title:: Part II. The theory of variables

Collection:: Economics Books

VIL.—AVERAGES. 127 form of average of the Y’s for any given year will afford an indication of the general level of prices for that year, provided the commodities chosen are sufficiently numerous and representative. The question is, what form of average to choose. If the geometric mean be chosen, and &,,, G,, denote the geometric means of the Y’s for the years 7 and 2 respectively, we have Go_(T Tn I's i Gy \Y') YI ; C5 2: SN ARR Se A LL - (Fy 7 BE) From the first form of this equation we see that the ratio of the geometric mean index-number in year 2 to that in year 7 is identical with the geometric mean of the ratios for the index- numbers of the several commodities. A similar property does not hold for any other form of average : the ratio of the arithmetic mean index-numbers is not the same as the arithmetic mean of the ratios, nor is the ratio of the medians the median of the ratios. From the second and third forms of the equation it appears further that the ratio of the geometric mean index- number in year 2 to that in year 7 is independent of the prices in the year first chosen as base (i.e. year 0), and is identical with the geometric mean of the index-numbers for year 2, on year 7 as base. Again, a similar property does not hold for any other form of average. If arithmetic means of the index-numbers be taken, for example, the ratio of the mean in year 2 to the mean in year 1 will vary with the year taken as base, and will differ more or less from the arithmetic mean ratio of the prices in year 2 to the prices of the same commodities in year 7 ; the same statement is true if medians be used. The results given by the use of the geometric mean possess, therefore, a certain consistency that is not exhibited if other forms of average are employed. It was used in a classical paper by Jevons (ref. 4), though not on quite the same grounds, but has never been at all generally employed. 26. The general use of the geometric mean has been suggested on another ground, namely, that the magnitudes of deviations appear, as a rule, to be dependent in some degree on the magni- tude of the average; thus the length of a mouse varies less than the stature of a man, and the height of a shrub less than that of a tree. Hence, it is argued, variations in such cases should be measured rather by their ratio to, than their difference from, the average ; and if this is done, the geometric mean is the natural average to use. If deviations be measured in this way, a

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