Researches on the mode of distribution of the constants of samples taken at random from a bivariate normal population

Abstract The theory of samples, whatever their size, has been largely developed of recent years. This development may be said to have followed two independent lines. In the first of these there is no limitation to the nature of the frequency in the sampled population; that population has been supposed to be known by its momental coefficient, and the aim has been to determine the momental coefficient of the population of samples, and to find the successive momental coefficient of these momental coefficients themselves. Thus we know completely the first four momental coefficients of the distribution of means, and of standard deviations of samples of any size large or small taken from a finite population with any law of distribution. These give us some general idea of how these means and standard deviation are libels to occur in practice. But the expressions are very lengthy, and in the case of tbs third and fourth moments investigators have been reduced to approximations, or to supposing the population sampled "infinite," i. e., to supposing an individual just drawn to be returned to the sampled population before the next drawing. Some recent researches, experimental and theoretical, seem to indicate that if the sample be not more of about one-fiftieth of the sampled population, it is,for practical purposes, indifferent whether we consider the population sampled finite or "infinite." The second form of investigation is to obtain, if possible, not the momental constants, but the actual frequency distributions of the various characters which describe the distributions of samples. Thus the distributions of the means of samples from populations lolloping certain types of frequency curves can now be written down in algebraical form. But thus far this method of inquiry has not been very fruitful, except in the case when the sampled population is supposed to follow a normal law and is considered infinite. In this case the distributions of the means, standard deviations, and the correlation coefficients have now been very completely studied. There are, however, other characters of samples, What we may- term "compound characters," that contribute essentially to our knowledge of sampling, and of which it is possible to obtain the theoretical distributions. Illustrations of tins will be given in the present memoir. It must be remembered, of course, that they- only- apply to samples from normal population; but these samples may be as small or as large as we please, and the results may be possibly tend to throw light on the corresponding distributions for samples in the case of non-normal sampled populations.