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. 311 unreliability of observed statistical results, and the term probable error is given to this quantity. It should be noted that the word “probable” is hardly used in its usual sense in this connection : the probable error is merely a quantity such that we may expect greater and less errors of simple sampling with about equal frequency, provided always that the distribution of errors is normal. On the whole, the use of the ‘probable error” has little advantage compared with the standard, and consequently little stress is laid on it in the present work ; but the term is in constant use, and the student must be familiar with it. It is true that the “ probable error ” has a simpler and more direct significance than the standard error, but this advantage is lost as soon as we come to deal with multiples of the probable error. Further, the best modern tables of the ordinates and area of the normal curve are given in terms of the standard-deviation or standard error, not in terms of the probable error, and the mul- tiplication of the former by 0:6745, to obtain the probable error, is not justified unless the distribution is normal. For very large samples the distribution is approximately normal, even though p and ¢ are unequal ; but this is not so for small samples, such as often occur in practice. In the case of small samples the use of the “probable error” is consequently of doubtful value, while the standard error retains its significance as a measure of dispersion. The ¢ probable error,” it may be mentioned, is often stated after an observed proportion with the + sign before it; a percentage given as 205 + 2-3 signifying “20'5 per cent., with a probable error of 2'3 per cent.” If an error or deviation in, say, a certain proportion p only just exceed the probable error, it is as likely as not to occur in simple sampling : if it exceed twice the probable error (in either direction), it is likely to occur as a deviation of simple sampling about 18 times in 100 trials—or the odds are about 4'6 to 1 against its occurring at any one trial. For a range of three times the probable error the odds are about 22 to 1, and for a range of four times the probable error 142 to 1. Until a deviation exceeds, then, 4 times the probable error, we cannot feel any great confidence that it is likely to be “significant.” Itis simpler to work with the standard error and take + 3 times the standard error as the critical range : for this range the odds are about 370 to 1 against such a devia- tion occurring in simple sampling at any one trial. 18. The following are a few miscellaneous examples of the use of the normal curve and the table of areas. Example i.—A hundred coins are thrown a number of times. How often approximately in 10,000 throws may (1) exactly 65 heads, (2) 65 heads or more, be expected §

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