Mann-Whitney compares means only when the distributions fulfill all assumptions required by the t-test.[#] If these assumptions do not hold, Mann-Whitney is usually a test on stochastic equality (that's a different thing!). The distinction is relevant only for "small" samples, since asymptotically stochastic equality and equality of means are the same.

So the question is: what hypothesis is it precisely you want to test? Stochastic equality or equality of the expected values? If it is the second, then the t-test is the only option. Knowing that the assumptions for this test are severely violated may be a sign to interpret the results with greater care, but there is no other test for this hypothesis.

Having massively non-normal residuals often is a result of a model mis-specification. Often one tries to infer differences (absolute changes) where it would be more sensible to infer ratios (relative changes)*. Having such "problems" like non-normal residuals should in the first place make one think about the model and the (sensibility of the) question one is trying to answer, and not which "alternative test" one can chose to generate some p-value (that 99% of the people wouldn't interpret correctly, if asked, anyway).

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*in this case the standard methods applied to the log-transfed response will do the job (but one could also consider a gamma-model)

[#] As Jorge pointed out below, this statement is exaggerated. In fact, the distributions just need to be identical (no matter what the form is) except for a location shift. In this case the location shift is theonly thing that can be and that is tested, and a location shift will equally affect the medians (like any other quantile as well!), the means (if they exist) and the mode (if it exists). There isnoneed that the distributions must be unimodal symmetric or even normal. However, I don't know any example of real distributions that differ only in their location (so I don't have an example where this interpretation would be applicable). Thank you, Jorge.

Exactly on point @Dr.Jochen Wilhelm

Hi! 

To be on a safer side,I think Mann-Whitney test would be a better option as in B data set  distribution is not normal.  

I've just finished a statistic course...if one of the two variable analysed follow a no-parametric distribution, a NO parametric test should be used as Mann-Whitney test.

Giacomo, this seems to having been an interesting statistics course ;)

What was the definition given for a "no(n)-parametric distribution of a variable"?

Hi Jochen,

They say that the variable follow a non-parametric distribution when the distribution is not "normal"; in other words, that the varieble analysed do not follow a normal gaussian distribution, which imply the fact that the mean and the median are not the same...but in order to assess so, one should rum a Kolmogorov-Smirnoff analysis that one can do through statistic program as SPSS....

Ok, that's a weird definition of "non-parametric". The correct term would rather be "non-normal". There are many other probability distribution models that are based on parameters of the data. Examples you might know are discrete distributions like the binomial and the Poisson distribution, and the exponential and uniform distribution as examples of continuous distributions. Then there are many very useful distributions like the log-normal, gamma, beta, chauchy, negative-binomial, weibull, rayleigh, dirichlet, wishart, ...

They all have sensible parameters for certain types of data. Understanding these distributions is actually essential to understand your data and to understand what we want (and what we can and what we should expect) from a statistical analysis. It seems that the course was not quite helpful in getting an idea of "statistical thinking".

A further note: the conclusion that non-normal distributions have mean ≠ median is wrong. All symmetric distributions with finite mean (what excludes the cauchy-distribution) have mean=median. The normal distribution is just one of those.

Giacomo, don't get me wrong, do not critisize you! I just want to open your eyes for the fact that the course you attended might have been far less useful than you think and that you missed the more important insights (not your fault!). Maybe I can motivate you to re-think these concepts, lokking "behind" these mechanic recipies provided in the course and obtain a better understanding about data, information, knowledge, probability, inference, and, eventually, science. What you learnt there to me looks more like a recipe to kill science.

Ádám Simon

I agree with Jochen Wilhelm. Sometimes, when I have situations like yours, I use the Van der Waerden's test, because as far as I know, it is a good alternative to the Student's test and to the analysis of variance when there are more than two samples. You can read

Haiko Luepsen (2018) Comparison of nonparametric analysis of variance methods: A vote for van der Waerden, Communications in Statistics - Simulation and Computation, 47: 9, 2547-2576, DOI: 10.1080 / 03610918.2017.1353613

The thing that I love about these discussions is that I frequently learn new insights. Thank you all and Jochen Wilhelm especially.

You can make the same transformation on the data of the two used variables. If the two transformed data have the normal distribution, you can use the t-test (parametric test) on the transformed data. If not (one or the two transformed data are not normally distributed), you must use a nonparametric test.

Ádám Simon

Taking a closer look at the conditions of your question, I suggest to apply a computer-intensive method, such as a permutations or bootstrapping test. I don't think a transformation will fix your problem.

Jochen Wilhelm, it seems to me that "Mann-Whitney compares means only when the distributions fulfill all assumptions required by the t-test" is exaggerated. Rank tests are applied in cases where the distributions of Xi and Yj have the same shape (F) and differ only in their central location. F is not even required to be symmetric. In these cases, we can well speak of the difference in means or medians and the Hodges-Lehmann method allows us to estimate it (1). When it is not specified that the effect is limited to a shift in the central location, the Mann-Whitney procedure usually loses the ability to compare these parameters and becomes a stochastic equality test, as you well point out. These are the reasons that lead me to suggest in Ádám Simon's case to apply a permutations or bootstrapping test.

(1) Hettmansperger, T., Statistical Inference based on ranks. John Wiley, 1984, p. 132...

Jorge Ortiz Pinilla , thank you for your comment on my statement. I absolutely agree with you. My statement was exaggerated. The shape of the distribution does not matter. It must only be the same in both groups, the location (shift) being the only difference. If the distributions have a mean, then the test tests a difference in means as well. And if the distributions have a mode, then the test tests a difference in modes as well.

I added a comment with a correction to the referred post.

I will advise testing the normality of the difference between A and B (A-B). In case that difference is normal (p>0.05), go for parametric test (t-test); in the contrary case (p

Thank you, Wilhelm

Depend on the scale used to measure the variable and the sample size and What scale you have used to measure the variable?

For normal distribution you should go for student's t test and for non-normal distribution go for Mann-Whitney. Now the question is how to compare!

Really waiting for the solution of your question from experts.

What needs to be considered here is to examine the assumption of normality. An independent normal t-test is required to assume an approximate normality. In this case, the approximate normality assumption for a large sample size (usually more than 50 or more than 100) is best checked with a Q-Q plot. The use of Shapiro-Wilk and Kolmogorov-Smirnov tests is not recommended for data with a sample size greater than 100 because they are sensitive to the slightest departure from normal.

Therefore, it can be done in the following order:

1. Check the sample size. If it is less than 50, Shapiro-Wilk can be a useful measure and test the normal distribution accordingly. If it is more than 50, the Q-Q plot can be used.

2. Once there is an approximate normality, the Independent t-test can be used.

3. The Mann-Whitney test can also be used in any situation.

4. We do not expect the results of these two tests to be different in the case of normal data distribution.

If you have a small sample size and data does not meet the assumption of normality, you should use the Mann-Whitney U test. Read more here https://www.reneshbedre.com/blog/mann-whitney-u-test.html

The approximate normal was needed in the t-test. For the purpose of evaluating the normal distribution, the QQ-plot was the best method to use. As long as the Severe deviation from normal was not observed, an independent t-test could be used to compare means. The Man-Whitney U test should be used to compare the distribution of numeric variables between groups if severe deviation was observed in the QQ-plot.