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Maybe I'm missing something, but it seems like the assumptions section is extremely wrong. The underlying distributions do *not* need to be normal. The statistics' (i.e., sample average) distributions need to be normally distributed, and they will be, according to the Central Limit Theorem. 70.35.57.149 (talk) 19:13, 7 March 2017 (UTC)[reply]
My understanding is that you are right, mostly. Only for small samples do we need the sample(s) to follow a normal distribution, when the mean (numerator) and standard error (denominator) won't automatically be normally distributed according to the CLT. And this is the situation where t-tests are most important, because when the samples are large enough for the CLT to apply, they're also large enough for the t-distribution to converge to the Z-distribution. I think this ought to be mentioned (although my authority for this is a statistician friend - I'm still looking for a published statement about it). Then the bit that describes how to test a sample for normality brings a special irony, because a test (like the Shapiro-Wilk or Kolmogorov-Smirnov) for normality is more likely to reject the null hypothesis of normality as the sample size becomes larger, and this is exactly when you don't need to worry so much about normality! RMGunton (talk) 15:45, 13 February 2019 (UTC)[reply]
The sample mean need not be normally distributed either. Sketch of proof: Efron (1969) (Student's t-Test Under Symmetry Conditions) shows in Section 1 that a proof by Fisher (1925) (Applications of "Student's" Distribution) for the normal case actually only uses the 'sphericity / rotational invariance / orthogonal invariance' of the normal distribution of individual observations for the t-test to control size (Type I error). So, orthogonal invariance of the distribution of X := (X_1, X_2, ..., X_n) is sufficient. This absolutely does not imply that the sample mean is normally distributed, so normality of the sample mean is not necessary. For (counter)example, if n = 3 then it follows from Archimedes' Hat-Box Theorem that a random variable distributed uniformly over the unit sphere (which is clearly orthogonal invariant) has a sample mean that follows a uniform distribution. NWK2 (talk) 14:31, 3 June 2021 (UTC)[reply]
I added a tag "dubious" to assumptions section. I agree that the distribution does not to be normal. I further think that variance does not have to follow Chi squared distribution. Even if part of it is true, it sounds very misleading. I included Shapiro-Wilk test in an official document before running the t-test, partly because of this Wikipedia page. 
Should these two assumptions be deleted entirely, or should one or both be substituted with some other statements in order to not be misleading? 38.104.28.226 (talk) 16:10, 17 October 2022 (UTC)[reply]
hey there, just wanted to point out that the values in the Worked_examples present some speed bumps for folks following along with some tools. in excel/google sheets terms, this is the difference in STDEV() versus STDEVP(). some tools, like numpy.std default to the latter so the values end up differing from examples. i will suggest an edit with values that avoid this that follows for the rest of the example, but wanted to flag this first.
along these lines, it is somewhat confusing that the difference in means just happens to `0.095`, something that is generally a value used for confidence thresholds. i think any suggestion to fix the first point will take care of this too, but a nice to have to avoid confusion for stats newbie's like me who'd be following this page topic. — Preceding unsigned comment added by StevenLinde (talk • contribs) 18:32, 22 May 2022 (UTC)[reply]
level-4 vital article is rated B-class on Wikipedia's content assessment scale.
