1) No. Formal tests on assumptions are nonsense. They don't ever "confirm" that your data follows a normal distribution. They can only tell you if your sample size is already large enough to recognize that your data is not following the (idealized!) normal distribution model.

2. See 1. These tests are made to let you know if your sample is large enough, so they will work with small samples as well, but this is all not the question that is relevant for you.

3. It is true that fromal tests are unneccesary. But it is neccesary that the biologist has a qualified idea what a reasonable distribution model is for his/her data and to use an appropriate statistical model and/or transformation.

4. No, not at all. You can rather savely assume that your data is not normal distributed. Only in few specail cases the normal distribution assumption may be reasonable, but typically it is not. In particular, the assumption is typically not reasonable for all variables cannot have negative values, like concentrations, intensities, proportions, percentages, counts, distances, masses, etc.

5. This question seem unrelated to the previous four. There is no simple answer; it depends pretty heavily on the purpose and the frequency of use. My requirements would be that such software is open, free, flexible and does not provide solutions that look like "one size fits it all" and "just click a button and everything will be done automatically". If I look at these requirements, "R" seems to be a good option (www.r-project.org). That's of course hard/tough, but it pays off a thousand times.

Let me add another way for you to think about Jochen Wilhelm's response to your first question. Presumably, the number of data points in your sample is finite. Given that a normal distribution is both continuous and infinite (min = -infinity, max = infinity), the data in your sample is definitely not normally distributed. ;-)

You mentioned ANOVA. ANOVA is a special case of the general linear model (GLM). For such models, the necessary normality assumption is that the sampling distributions of the parameter estimates are approximately normal.* For ANOVA, that means that the sampling distributions of the group means must be approximately normal. And the shapes of those sampling distributions depend on both the shapes of the underlying populations and the sample size(s). If the underlying populations were perfectly normal, then the sampling distributions would also be perfectly normal, irrespective of sample size. But as n increases, the sampling distributions approach normality even if the underlying populations are not normal.

As a rough rule of thumb, I suggest that if it is both sensible and defensible to use means and SDs for descriptive purposes, then ANOVA will probably be just fine. YMMV.

* I said approximately normal, because I believe George Box was right when he said that nothing in the real world is normally distributed. See the attached image.

Checking the normality of your data is important when using parametric statistical tests like t-tests and ANOVA. These tests make certain assumptions about the distribution of the data, and if those assumptions are violated, it can affect the validity of your results.

For t-tests and ANOVA, normality is particularly important when dealing with smaller sample sizes. However, these tests are somewhat robust against violations of normality, especially with larger sample sizes (typically, n > 30 is considered reasonably robust).

If normality is a concern, you might consider non-parametric tests like the Mann-Whitney U test (for two groups) or the Kruskal-Wallis test (for multiple groups) as alternatives.

In summary, while it's a good practice to check for normality, the impact of deviations from normality depends on your sample size and the specific assumptions of the statistical test you're using.

Miloud Chakit , you write:

"If normality is a concern, you might consider non-parametric tests like the Mann-Whitney U test (for two groups) or the Kruskal-Wallis test (for multiple groups) as alternatives."

I don't subscribe and I don't think that this is a good advice. These tests are about entirely different hypotheses, and these do not line up with the hypotheses tested by t-Tests/ANOVAs when the distributions are skewed with different variances! In such cases, it's not trivial to identify what makes a "location shift" (this is what the researcher typically wants to test):

Consider an example with one control group and one treated group, where the treatment is hoped to increase the measured response. It's possible to construct cases where the mean and median difference have opposite signs. Say in this example that the median difference is positive and that the mean difference is negative. What would you conclude from a significant MWU-test in this case? That the treatment is beneficial, because the median difference (treated - control) positive, or that the treatment is harmful, because the mean difference is negative?

Instead of advising "non-parametric statistics" as a general alternative, one should advice to understand the data-gererating process, to identify a reasonable (approximate) distribution model and to actually use this in the statistical analysis, and to test the relevant hypotheses within these models.

If such an advice seems to demanding, then the advice should be to collaborate with a statistician who has experience in dealing with "non-normal" data.

Thank you Professor Miloud Chakit

normality should be taking in consideration

In most cases it is important to check. This help to increase the validity in some cases reliability of the work. For instance, in the use of ANOVA be mindful that at the parametric assumptions that goes with this statistical tool is followed.

Thank you scholars for your answers and your insights and expertise on normality of data. However, after going through all the responses, I find myself aligning with Jochen Wilhelm . I appreciate everyone for sharing the wealth of knowledge and request you to give me more insights as I am new to this facet of research.

What do you think about transform Y for another scale lije Log(Y) and verify if its follow a Normal dist and run the t test?

BoxCox is a good way to find a power to transform Y.