This paper by Harold Hotelling discusses the generalization of Student's distribution to handle multiple variables and their simultaneous deviations. It begins by explaining the original Student's distribution, which is used to estimate the accuracy of a normally distributed quantity based on its variance. Fisher later extended this distribution to apply to regression coefficients and other quantities derived from least squares. The paper then introduces a generalization of this distribution to handle multiple variables. It defines a new statistic, T², which measures the simultaneous deviations of several quantities. This statistic is derived from linear functions of the observed values, ensuring that the variance is the same as for a single observation. The distribution of T² is shown to be invariant under linear transformations, making it suitable for various applications. The paper also discusses the application of this generalized distribution to problems in biology, such as determining whether two groups belong to the same race, and in other fields like economics, where the comparison of commodity prices over time is needed. It introduces a new coefficient of racial likeness, H, which is the average of the ratios of variances in two samples. However, H is not suitable for comparing magnitudes of characters, as it only considers variability. The paper further explains how the generalized distribution can be used to compare regression coefficients and slopes of secular trends. It also discusses the use of this distribution in testing the significance of deviations from hypothetical mean values, and in comparing the means of two samples. The paper concludes by showing that the generalized distribution can be used to determine the probability of a given value of T being exceeded by chance. It also discusses the use of the incomplete beta function to calculate this probability and the substitution of a logarithmic transformation to simplify the calculation. The paper emphasizes the simplicity and independence of the distribution of T from any correlations among the variables, making it a powerful tool for statistical analysis.This paper by Harold Hotelling discusses the generalization of Student's distribution to handle multiple variables and their simultaneous deviations. It begins by explaining the original Student's distribution, which is used to estimate the accuracy of a normally distributed quantity based on its variance. Fisher later extended this distribution to apply to regression coefficients and other quantities derived from least squares. The paper then introduces a generalization of this distribution to handle multiple variables. It defines a new statistic, T², which measures the simultaneous deviations of several quantities. This statistic is derived from linear functions of the observed values, ensuring that the variance is the same as for a single observation. The distribution of T² is shown to be invariant under linear transformations, making it suitable for various applications. The paper also discusses the application of this generalized distribution to problems in biology, such as determining whether two groups belong to the same race, and in other fields like economics, where the comparison of commodity prices over time is needed. It introduces a new coefficient of racial likeness, H, which is the average of the ratios of variances in two samples. However, H is not suitable for comparing magnitudes of characters, as it only considers variability. The paper further explains how the generalized distribution can be used to compare regression coefficients and slopes of secular trends. It also discusses the use of this distribution in testing the significance of deviations from hypothetical mean values, and in comparing the means of two samples. The paper concludes by showing that the generalized distribution can be used to determine the probability of a given value of T being exceeded by chance. It also discusses the use of the incomplete beta function to calculate this probability and the substitution of a logarithmic transformation to simplify the calculation. The paper emphasizes the simplicity and independence of the distribution of T from any correlations among the variables, making it a powerful tool for statistical analysis.