# Are normally distributed sample means equivalent to normally distributed residuals?

According to t-test:Assumptions "The means of the two populations being compared should follow normal distribution"

The one way ANOVA test in case of 2 groups equals the t-test but the ANOVA assumption on normality according to ANOVA:Assumptions is defined by "the distributions of the residuals are normal."

Question: Can I conclude that "means of sample distributions are normally distributed" is equivalent to "the distributions of the residuals are normal"?

The wikipedia link you give are using imprecise/confounded language, so you should maybe find some better tutorial! It also confuses assumptions and mathematical conclusions from those assumptions (it isn't an assumption that $$S^2$$ follows a $$\chi^2$$-distribution with $$n − 1$$ degrees of freedom, that can be concluded from the normal assumption and independence.) Find a better guide!
No, that is wrong. If you have a sample $$X_1,\dotsc,X_n$$ from some non-normal distribution, but the CLT applies and $$\bar{X}_n$$ is approximately normal, the residuals are $$X_i−\bar{X}_n$$, and its distribution is mostly that of $$X_i$$, non-normal (but recentered). The reason that the residual distribution form is mostly that of $$X_i$$, is that the mean has much lower variance.
• No, that is wrong. If you have a samle $X_1, \dotsc, X_n$ from some non-normal distribution, but the clt applies and $\bar{X}_n$ is approximately normal, the residuals are $X_i-\bar{X}_n$, and its distribution is mosly that of $X_i$, non-normal (but recetered). The reason that the residual distribution form is mostly that of $X_i$, is that the mean has much lower variance. Feb 14, 2020 at 17:33