What is the best statistical test for skewed distributions with different spread? (See image) I am working on an exercise in which I have to study how does one specific feature of a listing on a website for house rentals affects the number of reservations.    
I plotted my kernel density estimator (similar to a histogram) for the number of reservations for each variant of the feature:

I was thinking about doing a statistical test for the means, but since the distributions have such a different shape I am not sure if it is the best approach. Perhaps a statistical test for the medians would be more accurate?    
I checked the distribution for the means of each one of the populations by randomly picking sub-samples and computing the mean and it looks Gaussian, so I think a t-test should be fine. 
But since the shape of my distributions are clearly non-Gaussian, perhaps a non-parametric test would be a better choice?
 A: *

*Since your observations are counts -- discrete and with a hard (and attained!) lower bound of 0, I wouldn't just blithely treat them as continuous by using a kernel density estimator to smooth them. [The behavior below 0 is particularly an artifact of the KDE.]

*Allowing for the impact of the KDE, the shapes actually look somewhat similar to me (up to a different scale), but usually you wouldn't actually expect sets of counts to have the same shape (with the exception of samples from the geometric, which - at least sort of - does have the same shape). 

*In any case, if you want the same shape in order to use a nonparametric test, you'd only be looking for the same shape under the null; that is you don't necessarily expect the samples to look the same -- it's an assumption you'd want to hold when the null is true. The alternative can be more general in any number of ways where rejection would still be readily interpretable.

*Since these are counts I'd first be looking for a test that is more suited to count data. For example, I might consider a Poisson or Negative Binomial model (perhaps via GLM). 

*That's not to say you shouldn't consider a nonparametric test, though, so if that's your preference, that should be fine. If you want to test means, that can still be done with a nonparametric test (via a permutation test, for example). With small counts you'll have a lot of ties, though (e.g. if you were to do a Wilcoxon-Mann-Whitney, make sure your software is dealing with the ties, not just assuming these values are continuous).

*"I checked the distribution for the means of each one of the populations by randomly picking sub-samples and computing the mean and it looks Gaussian, so I think a t-test should be fine." -- it's not the distribution of the numerator, but the distribution of the t-statistic that matters. [Consider, for example, sample sizes of 30 from a Bernoulli(0.5). The numerator of a one-sample t-statistic looks fine, but because of the behavior of the denominator, the t-statistic itself is somewhat less like you'd expect it to be - the actual type I error rate is often quite different from the nominal rate across a large number of significance levels. e.g. at n=30, your 5% test is more like a 4.1% test, but at n=31 it's more like a 7.2% test!]
