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 III. Theory of sampling

Collection:: Economics Books

XVIL—SIMPLER CASES OF SAMPLING FOR VARIABLES. 353 “To convert any standard error to the probable error multiply by the constant 0-674489 , . . . 16. We need hardly restate once more the warnings given in Chap. XIV., and repeated in § 9 above, that a standard error can give no evidence as to the biassed or representative character of a sample, nor as to the magnitude of errors of observation, but we may, in conclusion, again emphasise the warnings given in §§ 1-3, Chap. XIV,, as to the use of standard errors when the number of observations in the sample is small. In the first place, if the sample be small, we cannot in general assume that the distribution of errors is approximately normal : it would only be normal in the case of the median (for which » and ¢ are equal) and in the case of the mean of a normal distribution. Consequently, if = be small, the rule that a range of three times the standard error includes the majority of the fluctuations of simple sampling of either sign does not strictly apply, and the “probable error” becomes of doubtful significance. Secondly, it will be noted that the values of o and Y, in (1), of Jn (2), and of o in (4) and (5), ie. the values that would be given for these constants by an indefinitely large sample drawn under the same conditions, or the values that they possess in the original record if the sample is unbiassed, are assumed to be known a priori. But this is only the case in dealing with the problems of artificial chance: in practical cases we have to use the values given us by the sample itself. If this sample is based on a considerable number of observations, the procedure is safe enough, but if it be only a small sample we may possibly mis- estimate the standard error to a serious extent. Following the procedure suggested in Chap. XIV., some rough idea as to the possible extent. of under-estimation or over-estimation may be obtained, e.g. in the case of the mean, by first working out the standard error of o on the assumption that the values for the necessary moments are correct, and then replacing o in the expression for the standard error of the mean by o + three times its standard error so obtained. Finally, it will be remembered that unless the number of observations is large, we cannot interpret the standard error of any constant in the inverse sense, 7.e. the standard error ceases to measure with reasonable accuracy the standard-deviation of true values of the constant round the observed value (Chap. XIV. § 3). If the sample be large, the direct and inverse standard errors are approximately the same. 23

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