Normality test and parametric vs non parametric tests

I'm pretty new in statistical testing and I'd like some help in what I'm doing. I have two groups of scores from 0-100. Group A has 10 values and group B has 70 values. Initially I'm testing normality using Shapiro-Wilk separately for each group. Is this correct or should I treat my data as one group with 80 values?

Assuming that is correct the separate testing, I found that Group A is not normal and group B is normal. Therefore I'm using the Mann-Whitney-Wilcoxon test for non-parametric testing as one of the groups is not normal. Is this correct or not?

After this I split Group B in 4 sub-groups with different sample sizes (11,20,24,15). Now I have 5 groups, Group A (same as initially) which is not normal and 4 sub-groups from the initial Group B which are normal again using Shapiro-Wilk.

Then I'm testing all 5 groups using Kruskal-Wallis for significant difference (non-parametric test, as I have one non-normal sample). From this I get significant difference among the five groups.

Finally using t-Test (when both samples are normal) and Mann-Whitney-Wilcoxon (when one of the two samples are not normal) I test all combinations of groups two by two in order to find which groups have and which don't have significant differences. Is this correct?

Could you help me with this? Any references to books or papers would be really helpful for me. I hope I used terminology correctly and explained my case well enough.

• Such scores in range 0-100 cannot have a normal distribution ... See stats.stackexchange.com/questions/2492/… Oct 19, 2022 at 18:03
• Nothing is ever normally distributed as real data are discrete and bounded. This however doesn't mean that a t-test can't be applied, as this is asymptotically valid (under assumptions) also for non-normal data. The use of normality tests here is quite controversial because neither does not rejecting mean that data are normal, nor does rejecting always mean that Wilcoxon will be better than the t-test. For a discussion see here: arxiv.org/abs/1908.02218 also @kjetilbhalvorsen Oct 19, 2022 at 18:16
• I'd look at the data first to see whether there are violations of normality that really cause problems (outliers, strong skewness, wildly different variances) rather than running a normality test. This is however hard to recommend to people without experience, as it is somewhat subjective. They could just run the Wilcoxon to be on the safe side, without any prior mis-specification testing. Better collaborate with an experiences statistician. Oct 19, 2022 at 18:20