Harold Hotelling's paper "The Generalization of Student's Ratio" discusses the extension of Student's distribution to handle more complex statistical problems. The paper begins by reviewing the work of "Student" and R. A. Fisher, who initially studied the ratio of the mean to the standard deviation of a sample. Hotelling then extends this concept to linear functions of normally distributed observations with equal variance, and introduces the measure \( T^2 \) for testing deviations from hypothetical values. The paper generalizes Student's ratio to handle cases where variances are not equal but have known ratios, and where observations are correlated. It also discusses the application of these methods to regression coefficients and other quantities obtained by least squares. Hotelling introduces the measure \( T^2 \) for simultaneous deviations of multiple quantities, which is a function of the sum of the squares of the deviations from the mean values. The paper further explores the geometric interpretation of \( T \) as the angle between a line and a flat space containing other lines drawn independently through the origin. This geometric approach helps in deriving the exact sampling distribution of \( T \). The distribution of \( T \) is expressed in terms of the incomplete beta function, and methods for calculating it are discussed, including the use of tables and approximations. Finally, Hotelling emphasizes the advantages of using \( T \) over other measures, such as the Pearson coefficient, due to its simplicity and independence from correlations among the variates. The paper concludes with a discussion on confidence intervals and the use of \( T \) in determining the true values of a set of measurements.Harold Hotelling's paper "The Generalization of Student's Ratio" discusses the extension of Student's distribution to handle more complex statistical problems. The paper begins by reviewing the work of "Student" and R. A. Fisher, who initially studied the ratio of the mean to the standard deviation of a sample. Hotelling then extends this concept to linear functions of normally distributed observations with equal variance, and introduces the measure \( T^2 \) for testing deviations from hypothetical values. The paper generalizes Student's ratio to handle cases where variances are not equal but have known ratios, and where observations are correlated. It also discusses the application of these methods to regression coefficients and other quantities obtained by least squares. Hotelling introduces the measure \( T^2 \) for simultaneous deviations of multiple quantities, which is a function of the sum of the squares of the deviations from the mean values. The paper further explores the geometric interpretation of \( T \) as the angle between a line and a flat space containing other lines drawn independently through the origin. This geometric approach helps in deriving the exact sampling distribution of \( T \). The distribution of \( T \) is expressed in terms of the incomplete beta function, and methods for calculating it are discussed, including the use of tables and approximations. Finally, Hotelling emphasizes the advantages of using \( T \) over other measures, such as the Pearson coefficient, due to its simplicity and independence from correlations among the variates. The paper concludes with a discussion on confidence intervals and the use of \( T \) in determining the true values of a set of measurements.