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

XL.—CORRELATION : MISCELLANEOUS THEOREMS. 221 often happens in the case of statements as to market prices), take the arithmetic mean (28s. 4d.) as the general average. But if we know that 23,930 qrs. were sold at 4, only 26 qrs. at B, and 3933 qrs. at C, it will be better to take the weighted mean (29s. 1d. x 23,930) + (27s. 7d. x 26) + (28s. 4d. x 3933) 99 ~ 27889 oT to the nearest penny. This is appreciably higher than the arithmetic mean price, which is lowered by the undue importance attached to the small markets B and C. In the case of index-numbers for exhibiting the changes in average prices from year to year (¢f. Chap. VIL. § 25), it may make a sensible difference whether we take the simple arithmetic mean of the index-numbers for different commodities in any one year as representing the price-level in that year, or weight the index-numbers for the several commodities according to their importance from some point of view ; and much has been written as to the weights to be chosen. If, for example, our standpoint be that of some average consumer, we may take as the weight for each commodity the sum which he spends on that commodity in an average year, so that the frequency of each commodity is taken as the number of shillings or pounds spent thereon instead of simply as unity. Rates or ratios like the birth-, death-, or marriage-rates of a country may be regarded as weighted means. For, treating the rate for simplicity as a fraction, and not as a rate per 1000 of the population, Birth-rate of whole country = el _ =(birth-rate in each district x population in that district) - S (population of each district) i.e. the rate for the whole country is the mean of the rates in the different districts, weighting each in proportion to its population. We use the weighted and unweighted means of such rates as illustrations in §17 below. 16. It is evident that any weighted mean will in general differ from the unweighted mean of the same quantities, and it is required to find an expression for this difference. If » be the correlation between weights and variables, o,, and o, the standard- deviations, and @ the mean weight, we have at once 3(W.X)=NMw+ro,o,), whe... M=M+ ro, “noe 15)

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