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In student t test to compare the means of two samples, whether the normal distribution of each sample is prehypothesis or not? As we know, t test is used for comparing two independent small samples with the size of less than 30 (in each sample). In small sample usually the statistically distribution of sample is not normal. If normality distribution is prehypothesis in t test, we have to transforme (such as square root) data before t test.

kjetil b halvorsen
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mohammad
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    Does this answer your question? [Normality of dependent variable = normality of residuals?](https://stats.stackexchange.com/questions/60410/normality-of-dependent-variable-normality-of-residuals) – kjetil b halvorsen Jan 12 '20 at 20:41
  • The distribution underlying a sample has *nothing* to do with the sample size. Do not confuse the distribution of a sample--its *empirical* distribution--with the distribution you use to model it! (*No* empirical distribution is Normal.) – whuber Jan 15 '20 at 20:33

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No you don't need to transform before the test. Research has shown that the t-test is quite reliable from departures of normality. In addition, it's appropriate for small sample sizes of e.g. 4, 6, 8, etc. The only thing you wouldn't want is a strong outlier(s), since averages and s.d. are used. For example, the average of the four numbers 1,2,3, and 1000000 is a little above 250000 = 1000006/4. So on the number line, the average of 250000 lies somewhere in outer space a 1/4 of the distance to 1000000, where there happens to be no data.

The non-parametric Mann-Whitney or Wilcoxon rank sum test can guard against outlier effects as they are rank-based. The general idea is that if an outlier is present and ranks are used, the average of the ranks 1,2,3,4 for the four values 1,2,3,1000000 is 2.5=10/4 -- which is not precisely what Mann-Whitney does, but is similar with evaluations for ties. The alternative hypothesis for Mann-Whitney is also that the two samples came from different distributions. But you still need some sort of central tendency for non-parametric tests.

For the t-test, you also have to know if the s.d. are equal between the two samples. If not, you can make an adjustment to the d.f. ($\nu=n_1 + n_2-2$), or use a Welch test.

Another important point is that parametric and non-parametric 2-sample tests for equality of means are for unimodal distributions of data, not multimodal with several large humps or spikes in distributions.

Mann-Whitney is also not exactly the same as Wilcoxon rank sum, since there is a slight variation in the numerator of their test statistics.

If you look at tail probabilities for the $t$-distribution (range of $\nu$) and standard normal, you'll see that the standard normal pretty much starts to approximate $t$ near samples sizes of $n \sim 120$. So I would use a Z-test for samples sizes above $n=120$, and call it a large sample test, and simply say the $t$ is for small samples.

Computationally, I like to use Z tests for large samples, since computing tail probabilities for the t-distribution is very expensive when compared with Z-scores tail probabilities.