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Methods to test for equality of two normal distributions

机译：测试两个正态分布是否相等的方法

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摘要

Statistical tests for two independent samples under the assumption of normality are applied routinely by most practitioners of statistics. Likewise, presumably each introductory course in statistics treats some statistical procedures for two independent normal samples. Often, the classical two-sample model with equal variances is introduced, emphasizing that a test for equality of the expected values is a test for equality of both distributions as well, which is the actual goal. In a second step, usually the assumption of equal variances is discarded. The two-sample t test with Welch correction and the F test for equality of variances are introduced. The first test is solely treated as a test for the equality of central location, as well as the second as a test for the equality of scatter. Typically, there is no discussion if and to which extent testing for equality of the underlying normal distributions is possible, which is quite unsatisfactorily regarding the motivation and treatment of the situation with equal variances. It is the aim of this article to investigate the problem of testing for equality of two normal distributions, and to do so using knowledge and methods adequate to statistical practitioners as well as to students in an introductory statistics course. The power of the different tests discussed in the article is examined empirically. Finally, we apply the tests to several real data sets to illustrate their performance. In particular, we consider several data sets arising from intelligence tests since there is a large body of research supporting the existence of sex differences in mean scores or in variability in specific cognitive abilities.
机译：大多数统计从业人员通常在正常性假设下对两个独立样本进行统计检验。同样，大概每个统计学入门课程都针对两个独立的正常样本处理一些统计学程序。通常，引入具有相等方差的经典两样本模型，强调对期望值的相等性的检验也对两种分布的相等性的检验，这是实际目标。在第二步中，通常放弃等方差的假设。介绍了采用韦尔奇校正的两样本t检验和方差相等的F检验。第一个测试仅被视为中心位置相等性的测试，第二个则被视为散射均匀性的测试。通常，没有讨论是否可能以及在何种程度上可以测试基本正态分布的相等性，这在方差相等的情况的动机和处理方面是非常不令人满意的。本文的目的是研究检验两个正态分布是否相等的问题，并使用适合统计从业人员以及统计入门课程的学生的知识和方法进行检验。本文中讨论的各种测试的功能均通过经验进行检验。最后，我们将测试应用于多个真实数据集以说明其性能。特别是，由于存在大量的研究支持均值得分或特定认知能力变异性方面的性别差异，因此我们考虑了来自智力测验的几个数据集。

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