Statistical tests for uneven groups We performed an experiment (usability testing), which had two independent samples, one group with 5 participants and the other with 24. Based on the analysis of the distribution of the data (Shapiro-Wilk test), we concluded that we operate with data where some measurements distributed normally and others do not confirm to normal distribution.
I was wondering which test would be the most appropriate for data that is normally distributed and which for the data that is not normally distributed in our case, since we compare a group with 5 participants with a group that has 24? 
We found T-Test (for normally distributed data) and Mann Whitney test (for data that is not normally distributed), however, we have two groups that are uneven and based on the conflicting information we are not sure if we can perform these tests.
Thank you for your answer! 
 A: A simple yet robust approach could be to compute the difference between medians and then compute 95% confidence interval of such statistic with bootstrap (percentile). Such confidence interval would be adequate for inference.
You can find here some useful references:
http://r.789695.n4.nabble.com/CI-for-the-median-difference-td4399508.html
http://r.789695.n4.nabble.com/CI-for-the-median-difference-td4399508.html
http://www.statalist.org/forums/forum/general-stata-discussion/general/564770-hypothesis-testing-for-bootstrapped-differences-in-medians-in-a-randomized-clinical-trial
Yet, the small samples (5 cases in one group), limits substantially external validity, even if your inferential estimates were quite precise (anyway very unlikely).
A: normal data, uneven sample sizes:
Welche's t-test expects normal distribution, but allows for uneven samples sizes (and unequal variance between groups). For your samples with normal distribution, but uneven sample sizes, this test will likely give the most power. 
https://en.wikipedia.org/wiki/Welch%27s_t_test
non-normal data, uneven sample sizes:
Welche's t-test is also suitable to use with ranked data, which converts the test to a non-parametric test. If you are heavily concerned about uneven sample sizes, this may be a good candidate for your samples with non-normal distribution. Here, you rank order transform the data before running Welche's t-test. 
That being said, the Mann Whitney test (ranksum) you've been using already, is non-parametric, and should not be biased by uneven sample sizes. And so you will likely find that Welche's ranked t-test closely approximates the results given by a ranksum.
However (as others have mentioned), you may have a larger issue to solve, in that your one sample size is just 5 participants. A group this small will require a substantial effect size (mean difference) in-order to be found to be significant. Before proceeding further, it may be worth conducting a power analysis, in order to determine how many participants would be required to detect your hypothesized effect. 
A: I suggest Approximate Randomization (AR). It applies to test statistics following any distribution and I read somewhere that it is evaluated to be either equally accurate or more accurate than other methods that assume certain distributions even when their assumptions are met. Another nice thing about AR is that it immediately spits out the $p$-value (no need to lookup distribution tables as in $t$-test).
