The article discusses the significance test for sphericity in a normal \( n \)-variate distribution, focusing on testing whether a sample from a normal population has equal variances and zero correlations. The test is particularly useful when the sample consists of \( N \) \( n \)-dimensional vectors with mutually perpendicular coordinate axes, but with an arbitrary orientation. The likelihood ratio criterion is derived to test the hypothesis of sphericity, and the distribution of the statistic \( L_{s n} \) is analyzed for \( n = 2 \) and \( n = 3 \). For \( n = 2 \), the distribution is explicitly given, while for \( n = 3 \), a Pearson curve is fitted to approximate the distribution. The article also provides significance levels for \( n = 3 \) using the incomplete beta-function tables and suggests approximations for other values of \( n \). Additionally, the article includes a simple sampling experiment on confidence intervals, comparing different approaches to estimating the range of a rectangular population using sample ranges, sample averages, and the largest sample value.The article discusses the significance test for sphericity in a normal \( n \)-variate distribution, focusing on testing whether a sample from a normal population has equal variances and zero correlations. The test is particularly useful when the sample consists of \( N \) \( n \)-dimensional vectors with mutually perpendicular coordinate axes, but with an arbitrary orientation. The likelihood ratio criterion is derived to test the hypothesis of sphericity, and the distribution of the statistic \( L_{s n} \) is analyzed for \( n = 2 \) and \( n = 3 \). For \( n = 2 \), the distribution is explicitly given, while for \( n = 3 \), a Pearson curve is fitted to approximate the distribution. The article also provides significance levels for \( n = 3 \) using the incomplete beta-function tables and suggests approximations for other values of \( n \). Additionally, the article includes a simple sampling experiment on confidence intervals, comparing different approaches to estimating the range of a rectangular population using sample ranges, sample averages, and the largest sample value.