The content discusses statistical tests for sphericity in a normal n-variate distribution. It introduces the concept of sphericity, where all variances are equal and all correlations are zero. The test involves comparing the likelihood ratio criterion for a spherical population to that of a general normal population. The likelihood ratio statistic is derived, and its distribution is analyzed for different values of n. For n=2, the distribution of the test statistic is determined, and for n=3, a Pearson-type curve is fitted to approximate the distribution. The test is shown to be sensitive when n is much larger than m, and the results are compared with other criteria. The paper also discusses the significance levels for the test and provides tables for reference. The authors acknowledge the contributions of others and reference several statistical papers. The content is part of a larger study on confidence intervals and statistical inference.The content discusses statistical tests for sphericity in a normal n-variate distribution. It introduces the concept of sphericity, where all variances are equal and all correlations are zero. The test involves comparing the likelihood ratio criterion for a spherical population to that of a general normal population. The likelihood ratio statistic is derived, and its distribution is analyzed for different values of n. For n=2, the distribution of the test statistic is determined, and for n=3, a Pearson-type curve is fitted to approximate the distribution. The test is shown to be sensitive when n is much larger than m, and the results are compared with other criteria. The paper also discusses the significance levels for the test and provides tables for reference. The authors acknowledge the contributions of others and reference several statistical papers. The content is part of a larger study on confidence intervals and statistical inference.