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On the Distribution of the Proper Fractions
Published online by Cambridge University Press: 15 September 2014
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In statistical work which deals with integral variates, the data frequently appear in the form of ratios, or unreduced proper fractions, e.g. sex- and fecundity-ratios; and to facilitate comparison these are arranged in classes, all the ratios falling within the same class being considered as equivalent. These classes must, as far as possible, contain an equal number of the ratios. Further, in certain fields the different kinds of ratios do not all occur with the same frequency, that is, one denominator will occur more frequently than another, without any reference to the number of fractions having this denominator.
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- Copyright © Royal Society of Edinburgh 1906
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page 116 note * Cf. Karl Pearson, “On the Inheritance of Fecundity in Thoroughbred Brood-mares,” Phil. Trans. A, vol. cxcii. pp. 294–296 (1898), where the evenness of various distributions of fecundity-ratios is discussed empirically. For this reference, and also for suggesting the problem, I am indebted to Mr David Heron, M.A., who is at present studying under Professor Pearson.
page 125 note * The only alternative is the vanishing of the discriminant of the system of equations. It is easily seen that this disct. is the determinant which corresponds to the left-hand upper quadrant of the normal distribution, omitting the first two rows and the middle and left-hand columns, in such a way that the elements corresponding to fractions occurring in only one class are 1, those corresponding to the limits ½, and the other elements 0. Whether the disct. vanishes or not, the equations are always satisfied by the above values.