Was Quetelet’s Average Man Normal?

Quetelet’s data on Scottish chest girths are analyzed with eight normality tests. In contrast to Quetelet’s conclusion that the data are fit well by what is now known as the normal distribution, six of eight normality tests provide strong evidence that the chest circumferences are not normally distributed. Using corrected chest circumferences from Stigler, the χ2 test no longer provides strong evidence against normality, but five commonly used normality tests do. The D’Agostino–Pearson K2 and Jarque–Bera tests, based only on skewness and kurtosis, find that both Quetelet’s original data and the Stigler-corrected data are consistent with the hypothesis of normality. The major reason causing most normality tests to produce low p-values, indicating that Quetelet’s data are not normally distributed, is that the chest circumferences were reported in whole inches and rounding of large numbers of observations can produce many tied values that strongly affect most normality tests. Users should be cautious using many standard normality tests if data have ties, are rounded, and the ratio of the standard deviation to rounding interval is small.本研究采用八种正态性检验（normality tests）对凯特勒（Quetelet）的苏格兰胸围数据进行了分析。与凯特勒的结论相悖——后者认为该数据可由如今所称的正态分布（normal distribution）良好拟合——但八种正态性检验中有六种均提供了强有力的证据，表明胸围数据并不服从正态分布。采用斯蒂格勒（Stigler）修正后的胸围数据后，卡方（χ2）检验不再能提供反对正态性的强证据，但其余五种常用正态性检验仍可得出该结论。仅基于偏度（skewness）与峰度（kurtosis）构建的D’Agostino–Pearson K2检验与Jarque–Bera检验则显示，无论是凯特勒的原始数据还是经斯蒂格勒修正后的数据，均与正态性假设相符。导致多数正态性检验得到低p值（p-value）、即表明凯特勒的数据不服从正态分布的主要原因在于：胸围数据均以整数英寸为单位报告，对大量观测值进行取整操作会产生大量结值（tied values），这类结值会对绝大多数正态性检验造成显著影响。若数据存在结值、经取整处理，且标准差（standard deviation）与取整区间的比值较小时，使用者在使用多数标准正态性检验时应保持谨慎。