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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Chapter

Document type:: Monograph

Structure type:: Chapter

Title:: Part III. Theory of sampling

Collection:: Economics Books

XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 299 as before, and the four streams that result will bear the propor- tions ¢%: 39% : 3gp?: p®. The final set, at the heads of the vertical strips, will give the streams proportions ¢*: 4¢3%p : 69%? : 49p®: p*, and these streams will accumulate between the strips and give a r~presentation of the binomial by a kind of histogram, as shown. Of course as many rows of wedges may be provided as may be desired. This kind of apparatus was originally devised by Sir Francis Galton (ref. 1) in a form that gives roughly the symmetrical binomial, a stream of shot being allowed to fall through rows of nails, and the resultant streams being collected in partitioned spaces. The apparatus was generalised by Professor Pearson, who used rows of wedges. fixed to movable slides, so that they could be adjusted to give any ratio of g:p. (Ref. 13.) 6. The values of the mean and standard-deviation of a binomial distribution may be found from the terms of the series directly, as well as by the method of Chap. XIII. (the calculation was in fact given as an exercise in Question 8, Chap. VII., and Question 6, Chap. VIIL). Arrange the terms under each other as in col. 1 below, and treat the problem as if it were an arith- metical example, taking the arbitrary origin at 0 successes: as XV is a factor all through, it may be omitted for convenience. (1) oh (4) Frequency f. Dev. &. JE qn Te es —y nglp nq" 1p nq" 1p -1 i 3 — n(n —-1)g—2p? 2n(n —- 1)g"—2p? mn-1)n-2) | n(n—1)(n-2) 3n(n—-1)(n-2 23 7 Ml fy peg BEE D The sum of col. 1 is of course unity, i.e. we are treating IV as unity, and the mean is therefore given by the sum of the terms in col. (3). But this sum is n—-1)(n-2 np | "+ (n- Nye Tai) 1 Js 4 Sot ; =np(q +p)" =np. That is, the mean J/ is mp, as by the method of Chap. XIII

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