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

XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 309 expected theoretical frequency in a certain interval is f, the standard error of sampling is /A(N —f)/N ; and if the divergence of the observed from the theoretical frequency exceed some three times this standard error, the divergence is unlikely to have occurred as a mere fluctuation of sampling. It should be noted, however, that the ordinate of the normal curve at the middle of an interval does not give accurately the area of that interval, or the number of observations within it: it would only do so if the curve were sensibly straight. To deal strictly with problems as to fluctuations of sampling in the frequencies of single intervals or groups of intervals, we require, accordingly, some convenient means of obtaining the number of observations, in a given normal distribution, lying between any two values of the variable. 16. If an ordinate be erected at a distance z/o from the mean, in a normal curve, it divides the whole area into two parts, the ratio of which is evidently, from the mode of construction of the curve, independent of the values of y, and of o. The calculation of these fractions of area for given values of z/s, though a long and tedious matter, can thus be done once for all, and a table giving the results is useful for the purpose suggested in § 15 and in many other ways. References to complete tables are cited at the end of this work (list of tables, pp. 357-8), the short table below being given only for illustrative purposes. The table shows the greater fraction of the area lying on one side of any given ordinate ; e.g. 0'53983 of the whole area lies on one side of an ordinate at 0-1c from the mean, and 046017 on the other side. It will be seen that an ordinate drawn at a distance from the mean equal to the standard-deviation cuts off some 16 per cent. of the whole area on one side ; some 68 per cent. of the area will therefore be contained between ordinates at +o. An ordinate at twice the standard-deviation cuts off only 2:3 per cent., and therefore some 954 per cent. of the whole area lies within a range of +20. At three times the standard-deviation the fraction of area cut off is reduced to 135 parts in 100,000, leaving 997 per cent. within a range of +30. This is the basis of our rough rule that a range of 6 times the standard-deviation will in general include the great bulk of the observations: the rule is founded on, and is only strictly true for, the normal distribution. For other forms of distribution it need not hold good, though experience suggests that it more often holds than not. The binomial distribution, especially if p and ¢ be unequal, only becomes approximately normal when 7 is large, and this limitation must be remembered in applying the table given, or similar more complete tables, to cases in which the distribution is strictly binomial.

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