L'évolution industrielle de la Belgique

An Introduction to the theory of statistics

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Monograph

Identifikator:: 1029903786

URN:: urn:nbn:de:zbw-retromon-63471

Document type:: Monograph

Author:: Lewiński, Jan S. http://d-nb.info/gnd/102701059

Title:: L'évolution industrielle de la Belgique

Place of publication:: Bruxelles

Publisher:: Misch & Thron, Éditeurs

Year of publication:: 1911

Scope:: 1 Online-Ressource (XIII, 444 Seiten)

Digitisation:: 2018

Collection:: Economics Books

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Document type:: Monograph

Structure type:: Chapter

Title:: Première partie

Collection:: Economics Books

XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 313 Example iv.—The diagram of fig. 49 shows that the number of statures recorded in the group “62 in. and less than 63” is markedly less than the theoretical value. Could such a difference occur owing to fluctuations of simple sampling; and if so, how often might it happen ? The actual frequency recorded is 169. To obtain the theoreti- cal frequency we may either take it as given roughly by the ordinate in the centre of the interval, or, better, use the integral table. Remembering that statures were only recorded to the nearest % in., the true limits of the interval are 6115-6212 or 61:94-62'94, mid-value 62:44. This is a deviation from the mean (67°46) of 5°02. Calculating the ordinate of the normal curve directly we find the frequency 197-8. This is certainly, as is evident from the form of the curve, a little too small. The interval actually lies between deviations of 4:52 in. and 552 in., that is, 17590 and 2:1480. The corresponding fractions of area are 0'96071 and 0-98418, difference, or fraction of area between the two ordinates, 0:02347. Multiplying this by the whole number of observations (8585) we have the theoretical frequency 201-5. The difference of theoretical and observed frequencies is therefore 32:5. But the proportion of observations which should fall into the given class is 0023, the proportion falling into other classes 0-977, and the standard error of the class frequency is accordingly 0-023 x 0977 x 8585 =14'0. As the actual deviation is only 2:32 times this, it could certainly have occurred as a fluctuation of sampling. The question how often it might have occurred can only be answered if we assume the distribution of fluctuations of sampling to be approximately normal. It is true that 2 and gq are very unequal, but then =z is very large (8585)—so large that the difference of the chances is fairly small compared with npg (about one-fifteenth). Hence we may take the distribution of errors as roughly normal to a first approximation, though a first approximation only. The tables give 0-990 of the area below a deviation of 232s, so we would expect an equal or greater deficiency to occur about 10 times in 1000 trials, or once in a hundred. REFERENCES. The Binomial Machine. (1) GavroN, FraNcis, Natural Inheritance ; Macmillan & Co. London, 1889, (Mechanical method of forming a binomial or normal distribution, SREP ya p. 63; for Pearson’s generalised machine, see below, rel. .

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