data visualization
I am comparing the distrubution of different groups using the violin plots, however most of the online resources I found are just related to how to make the plots and very basic interpretation of the results (the median variation, the data is clustered or not).
I am looking for detailed examples that I can follow as my guideline to correctly interpret the violin plots.
Box plots are used for schematic summaries of a distribution. The violin plots are just box plots in which the Q1, Q2, and Q3 boxes are replaced by a wide range of quantiles. For that reason, I think the accepted practice is to use uniform width across groups.
However, you bring up a good point: how should densities across groups be compared? The answer depends on whether you are looking at each group as it's own population or as subpopulations.
I think that a useful DEFAULT behavior is to think of the full data as being the density we want to estimate. The groups are subpopulations such that the full density is a MIXTURE of the sub-densities. That suggests that each sub-density should be weighted by the number of observations. The areas (integral of the densities) of the k groups should be P_i, where $\Sigma_i P_i = 1$. This says that "Weighted Areas" is a good approach.
One example I enjoy (and is a simple illustration) is the work by Michael Maltz on analyzing the uniform crime reports that police agencies supply to the FBI. See:
Maltz, M. D. (2010). Look before you analyze: Visualizing data in criminal justice. In Piquero, A. . and Weisburd, D., editors, Handbook of Quantitative Criminology, chapter 3, pages 25-52. Springer New York, New York, NY.
For some background, the FBI does not have standardized ways to report missing or incomplete reports (they collect data monthly, so an agency could report for some months but not the entire year). So the uncritical would observe zeroes or very low numbers for a particular jurisdiction and not presume missing data, e.g. see the numbers for Florida in Parker & Pruitt (2000). So there is quite a bit of precedent in the criminology literature of modelling this data without discovering such errors.
Here is a good example from blogs discussing published papers:
There are also some good examples on this site. I thought I had a good example here but a few others that I really enjoyed are:
I realize these aren't published, but I think are illustrative nonetheless. I'm sure you could cull up more on this site as well.
Best Answer
A violin plot is just a histogram (or more often a smoothed variant like a kernel density) turned on its side and mirrored. Any textbook that teaches you how to interpret histograms should give you the intuition you seek. Edit per Nick Cox's suggestion: Freedman, Pisani, Purves, Statistics covers histograms.
As far as interpreting them in a more formal way, the whole point of graphing the distribution is to see things that statistical tests might be fooled by.
One thing I like to do with violin plots is add lines for the median, mean, etc. Sometimes I'll superimpose a boxplot so I can see even more in the way of summary statistics.
At very least, you should be able to pick out any gross deviations in the first few moments (mean, dispersion, skewness, kurtosis) as well as bimodality and outliers.