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. 347 of n, we may take the following case. If the student will turn to the calculated binomials, given as illustrations of the forms of binomial distributions in Chap. XV. § 3, he will find there the distribution of the number of successes for twenty events when ¢=09, p=0-1: the distribution is extremely skew, starting at zero, rising to high frequencies for 1 and 2 successes, and thence tailing off to 20 cases of 7 successes in 10,000 throws, 4 cases of 8 successes and 1 case of 9 successes. But now find the distribu- tion for the mean number of successes in groups of five throws, under the same conditions. This will be equivalent to finding the distribution of the number of successes for 100 such events, and then dividing the observed number of successes by five—the last process making no difference to the form of the distribution, but only to its scale. But the distribution of the number of successes for 100 events when ¢=09, p=0-1, is also given in Chap. XV. § 3, and it will be seen that, while it is appreciably asymmetrical, the divergence from symmetry is comparatively small : the distribution has gained very greatly in symmetry though only five observations have been taken to the sample. We may therefore reasonably assume, if our sample is large, that the distribution of means is approximately a normal dis- tribution, and we may calculate, on that assumption, the fre- quency with which any given deviation from a theoretical value or a value observed in some other series, in an observed mean, will arise from fluctuations of simple sampling alone. The warning is necessary, however, that the approach to normality is only rapid if the condition that the several drawings for each sample shall be independent is strictly fulfilled. 1f the observations are not independent, but are to some extent positively correlated with each other, even a fairly large sample may con- tinue to reflect any asymmetry existing in the original distribution {¢f. ref. 32 and the record of sampling there cited). If the original distribution be normal, the distribution of means, even of smali samples, is strictly normal. This follows at once from the fact that any linear function of normally distributed variables is itself normally distributed (Chap. XVI. § 6). The distribution will not in general, however, be normal if the deviation of the mean of each sample is expressed in terms of the standard-deviation of that sample (cf. ref. 30). 14. Let us consider briefly the effect on the standard error of the mean if the conditions of simple sampling as laid down in § 2 cease to apply. (a) If we do not draw from the same record all the time, but first draw a series of samples from one record, then another series from another record with a somewhat different mean and

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