Pairwise non-parametric test?

I have 5 groups of patients (different sizes), for which I measured some property that is continuous but not normally distributed. I would like to compare every pair of groups in respect to this property and get a p value, in order to determine which pairs are different from each other. What is the appropriate way to perform this?

Thank you!

And here's a plot to illustrate the distributions:

• Perhaps use a Kruskal-Wallis test at level $\alpha$ to see if there are any significant differences. If so perform ad hoc ${5\choose 2} = 10$ two-sample Wilcoxon tests at level $\alpha/10$ (Bonferroni criterion) to identify which pairs of groups differ. // It is difficult to give optimal advice based on such a skeletal and abstract description of the problem. In particular, it would be helpful to know whether the 'non-normal' property has the same type of distribution in all groups and about how large the 'different sizes' of groups are. – BruceET Dec 9 '18 at 18:41
• Thanks BruceET, a Kruskal-Wallis test results gave p value = 2.61979e-20. I also added a box plot illustrating the distribution as an edit to the question. A wilcoxon test with alpha=0.005 resulted in 7/10 significant pairwise differences. Is a Bonferroni preferable to FDR in this context? Thanks! – R Sorek Dec 10 '18 at 7:58
• In statistical software, 'notches' in sides of boxplots are often nonparametric CIs calibrated for comparing two plots at a time. I see indications there of about 7 signif differences. Might look to see if those agree with Bonferroni. Boxplots carry no inherent info about sample sizes; for publication these should always be reported in caption or nearby text. // Acronym 'FDR' doesn't immediately ring a bell; what's that? // Anyhow glad you're getting answers that make sense to you. – BruceET Dec 10 '18 at 17:51