# Which distributions are valid for Student's t-test?

When I read Wikipedia - Student's t-test I can see it assumes that:

1. The sample mean follows a Gaussian distribution.
2. The sampled mean follows a chi square distribution.

Samples of Gaussian distribution clearly holds the assumptions.
Are there any other distributions which obeys it?

While t-test when the data population is not normally distributed touches the subject, it only shows a single case. The question is about distributions which obeys the assumption on small number of samples or better yet arbitrary number of samples.

• From your link I found this stats.stackexchange.com/a/466368/350328 which shows when exponential distribution gets close to hold. I wonder if there's a family of such cases.
– Mark
Dec 28, 2022 at 14:23
• @User1865345, the asymptotic case via the CLT is, I'd say, well-understood. I think the more interesting question is whether, in finite samples, normality is necessary for the t-statistic to be an independent ratio of a normal and a chi-square r.v. I believe it is, but struggle to find a reference. Dec 28, 2022 at 14:31
• It might be typical for us to get questions on here to which an appropriate response is that the t-test is quite robust to many deviations from the assumption of a normal population distribution. This question, however, seems different by asking if there are other population distributions for which there is a theoretical guarantee of the t-stat having a t-distribution the way that $iiid$ normal data guarantee this. Count me as someone curious about this (despite, or maybe because of, my skepticism that another distribution can give this kind of guarantee).
– Dave
Dec 28, 2022 at 14:53
• @ChristophHanck - I may have misunderstood your comment, but normality of the sample mean characterizes the Normal distribution, as stated in Johnson, Kotz, and Balakrishnan (2nd edition.) This would appear to answer your question with a "yes". I cannot find an equivalent statement about the $\chi^2$ part, though I've found one that's close. Dec 29, 2022 at 1:37
• I’ve voted to reopen; I stand by my earlier comment in wondering if a t-distributed test statistic can be guaranteed for finite samples from a distribution that is not normal, and the link addresses robustness, not this theoretical guarantee.
– Dave
Jan 2 at 10:07