First problem: I assume that you have categorized 50% as "normally distributed" because the test for normality failed to be significant at the 0.05 level. The problem is that a failure to reject the null hypothesis is not the same as proving that the null hypothesis is true. My guess is that if you increase your sample size, there will be a point where all the variables will be non-normal.

I would try 1 and 2. I would also work at developing 3, 4, and 5. If all five approaches are valid (approach has few assumptions, different assumptions, is robust to departures from assumptions, or transformations alleviate problems), then I would ask how the outcomes differ qualitatively. If you get the same answer no matter which approach is chosen, that I would have great confidence in the answer. Choose the simplest approach and mention the other approaches that were tried in the methods. If they all give qualitatively different answers, then present your favorite answer and the answer that is as far from that as possible. Discuss. Alternatively, find a metric that will enable you to choose a model.

You have recorded different parameters of an insect species in the laboratory. This suggests that you have either dietary changes or insecticide applications and you have measured variables like time to pupation, body weight, fecundity, and the like. Try changing the problem if you can. None of these variables are discrete.

Consider a plant experiment where yield is a function of nitrogen fertilization. With two levels of nitrogen (2 kg NH4/ha and 200 kg NO3/ha) I have a discrete model because I have no confidence that the relationship is a straight line. If I test 0, 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024 kg NH4/ha, plus the same rates for NO3 (22 treatments in total), I still have a discrete treatment list but I am much better off using a regression analysis approach to looking at the effect of nitrogen on yield.

Dear Qasim,

1. the proper use of ANOVA is when: the variables are continuous; independence between the mean and variance; normality of the variables; homogeneity of the variables; you can inference between the sample and the population; n > 30 (in my personal experience is better n > 100).

2. KW tests the equality of medians of different groups; this method is the corresponding non-parametric analysis of variance, in which the data are replaced by their rank and is usually used when it can not be assumed a normal distribution of the population.

I hope to be helpful. Good job

Here are a few comments.

1. As George Box famously noted, nothing in nature is really normally distributed.

"In applying mathematics to subjects such as physics or statistics we make tentative assumptions about the real world which we know are false but which we believe may be useful nonetheless. The physicist knows that particles have mass and yet certain results, approximating what really happens, may be derived from the assumption that they do not. Equally, the statistician knows, for example, that in nature there never was a normal distribution, there never was a straight line, yet with normal and linear assumptions, known to be false, he can often derive results which match, to a useful approximation, those found in the real world."  Source: Box GEP (1976). Science and Statistics, Journal of the American Statistical Association, Vol. 71, No. 356, pp. 791-799. (Emphasis added.  Article available at the link below)

2. Testing for normality as a precondition to using a parametric test is not a very sensible thing to do, because the test of normality is under-powered when samples are small (and approximate normality is most important), and over-powered when samples are large (and approximate normality is less and less important, because the sampling distribution of the statistic converges on normality as n increases).  (I once gave a conference presentation on this issue--see one of the links below for the slides.)

3. Rather than testing for normality prior to using a parametric test, I think it is far more sensible to ask if you can fairly and honestly use means and SDs for description.  If you can, the parametric test is probably not going to lead you astray.

4. The assumptions for rank-based tests, given the way most people use them*, are arguably more onerous than the assumptions for parametric tests.  See the nice simulation study by Fagerland & Sandvik, for example. 

* See Fagerland & Sandvik's discussion of the "pure-shift" model in connection with the Wilcoxon-Mann-Whitney test, for example. 

HTH.

http://www-sop.inria.fr/members/Ian.Jermyn/philosophy/writings/Boxonmaths.pdf

http://www.nosm.ca/uploadedFiles/Research/Northern_Health_Research_Conference/Weaver,%20Bruce_Silly%20or%20Pointless%20Things.pdf

Article The Wilcoxon-Mann-Whitney test under scrutiny

In practical when your data show deviation from normality you can transform data to get normal distribution , then you can use parametric tests , in other hand when your data still not in normal distribution you can use nonparametric test .

Good Luck

I indead agree with and like the answers of Timoty and Bruce. Try several approaches and compare the results. You can also try permutation tests or more simpler (but robust) aproach like estimating confidence intervals by bootstrapping and vissually compare if and how much they overlap. Note that statistics is a tool not the aim of the study, so you always have to use logical, biological, ecological, and other meanings when drawing conclusions. And discuss your results critically as much as possible.

Best, Michal