# 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!

As to the question, does normal distribution of the sample means imply normal distribution of the residuals? If the sampling distribution of the sample means are exactly normal, then the residuals will be normal. But, if you only have some approximate normal sampling distribution of the sample means, then that conclusion do not follow.

Your extra question in comment: Can I argue as follows: the central limit theorem states that if the sample size is large enough, the distribution of the means is close enough to the normal distribution and can I therefore assume that the residuals are normally distributed?

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.

• Can I argue as follows: the central limit theorem states that if the sample size is large enough, the distribution of the means is close enough to the normal distribution and can I therefore assume that the rediduals are normally distributed? – methus Feb 14 '20 at 15:21
• 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. – kjetil b halvorsen Feb 14 '20 at 17:33