One-Way ANOVA with Non-Normal Distribution

@fausto.galetto  Since you were not able to reject the null hypothesis and see a difference, I would want to know the Power of your tests.  Given the small sample sizes is there no difference as you state or is the lack of noticeable difference due to a low power?

Not sure I understand what values you put into Minitab for difference in means.

I think I found that 10+10 items should give at least 90% power for the differences to be assessed

P-value for normality for Sample 1 is .485 and for Sample 2 is .020.  Close enough with respect to the normality assumption.  In that case, maybe the 2 sample t can be used.  The p values for the equal variances are just a bit above .05 so maybe close enough.  Bottom line is that maybe the 2 sample will work in this case.  Since no difference was seen and the sample size is small there is still a possible issue with Power.

The p-value for the 2 sample t is .092 which is a marginal conclusion of no difference.  The mathematical difference between the two means is 615.  I ran a Power and Sample Size calculation for a difference of 600 and the power turned out of be .38, not too strong.

Bottom line here is that the data sets are probably too small to be of any use in answering the original question of whether there is  any statistically significant differences between the two samples.  Run everything with an alpha of .10 and conclusions will probably change.

Then you can see above that the power for your data is only 38% which means if there is a difference you only have a 38% chance of seeing it.  With a sample size of only 10, there will be a significant overlap of the two confidence intervals so it will not be easy to see a difference.  I would not make a decision that the two groups are statistically the same with such a low power value.  To achieve a minimally acceptable power of 80% you will need at least 27 data points for each group.  Then you can run the test again and see whether you have a difference or not.