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There are several posts about interpreting violin plots like Violin plots interpretation or https://mode.com/blog/violin-plot-examples ,which I think I understand.

In my mind, however, I am asking a different question that has not been answered - what formal "statistics" can we use to determine if two groups, presented by violin plots, are the same or different.

For any two groups represented by violin plots, we get median, interquartile ranges and probability/density of being at certain values, ... etc, which are all good and well. But formally, how big a difference in the density plot, or in the interquartile ranges, or in the median, do they have to be for two groups to be considered different ? In the end, do we resort back to the traditional stats, such as Kruskal–Wallis test, t-test, to tackle each and every feature presented in the violin plots or are there formal statistics that can compare and contrast multiple features of the violin plots and conclude if the two groups are "statistically different" or not ?

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    $\begingroup$ Why and how are you comparing violin plots? These are just plots. Presumably you want to compare the data that was used to generate the plot? Or do you only have the data presented in the plot? $\endgroup$ Sep 16, 2019 at 5:57

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"Statistically different" doesn't mean anything by itself. You have to say statistically different with regard to what. So, yes, you might want to test means, medians and so on. Or you might want to test something else - if so, tell us what. It might be some aspect of the plot that is not captured by the usual statistics.

But you can't just test "are these plots different?"

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