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Note on the Standard Deviations of Samples of Two or Three

ANDREW W. YOUNG

1916
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Biometrika
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In an " Editorial" contained in VoL x of Biomebrika*, there is a discussion of the distribution of the values of the standard deviatkm of a population which aro deduced from small samples of the population. It is there shown how the distribution approaches normality as the number, n. in the sample increases, a table of the oharaoteristio oonstanta of the frequency carves for various value* of » being given. The smallest sample oonaidered is that of » = 4, but samples of two and three are of
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... sional occurrence especially in physical work and now and again a value of the probable error of an experimental result is deduced from a set of two or of three observations. A knowledge of the theoretical distribution of the standard deviations for such small mnpl« will give us some idea of the reliability of this procedure and it is the object of this note to supply this omission from the former paper. "Student's" formula for the distribution of samples of standard deviation is y where a is the standard deviation of the wholo population and 3 is the standard deviation givon by a sample of size a. Thus the distribution for satnplfis of two is extending from 2 = 0 to I = oo, Le. the distribution is simply half of a normal curve; and tho distribution for samples of three is also, of oourse, extending from 2 = 0 to 2 = oo. It in easy to find by direct integration the moment coefficients of these curvw. Coat of Sample* of Tvo. If we denote by N the total number of samples in the assumed distribution. ,\ Taking the first four moment coefficients about the origin to be p,', p,', p,', /*,', and the moment coefficients about the mean to be, as usual, m, pj, p*, we have -^ ( -Mean value, of J = i -^ (ze'^dl = ^ »^ = -5642<r. giving: w = and the standard deviation of 3 = o-j = -4263a-. * "On the Distribution of the Standard Deviations of Small Samplea." Appendix I to papers by "Stodent" and R. A. Hiher. BitmutrOea, VoL x. p. 822, 1915. at University of California, San Diego on June 12, 2015 http://biomet.oxfordjournals.org/ Downloaded from By this means the theoretical frequencies given in column (a) of Table £LT were calculated. A histogram of these values win be found to be well in accord with the half-Gaussian given above. • Pearson, P»i Trtau.

doi:10.1093/biomet/11.3.277
fatcat:gpeq4gt5dra77omeoqtqxckjv4