III. On the application of the theory of error to cases of normal distribution and normal correlation

Abstract

In his ‘Lettres sur la Théorie des Probabilitiés’ (1846), Quetelet has shown that in certain anthropometrical statistics, e. g. , in statistics of height or of chest-measurement, the curve of frequency is approximately of the same form as the curve known to mathematicians as the “curve of error,” but better described for statistical purposes as the normal curve . A similar conclusion has been arrived at by later observers with regard to a large number of biological measurements. The general similarity thus established has been extended, primarily by Mr. Francis Galton, to certain cases of statistical correlation of two or more attributes. It has been found in these cases that not only are the curves of frequency of the separate attributes approximately normal curves, but the frequencies of joint occurrence of different measures of these attributes follow (approximately) a simple law, corresponding to the law of correlation of errors of observation. Since we can never observe more than a finite number of individuals, it is impossible to decide with absolute certainty as to the existence, in any particular case, of this (or any other) law of distribution or correlation. But if the number of observed individuals is large, and if they are obtained by random selection from a “community” comprising (practically) an indefinitely great number of individuals, the theory of error provides us with a test for deciding whether any particular law, suggested by the given observations, may be regarded as holding for the original community.