# Can I say that my samples are different just by looking at box plots without performing a test?

Is it possible to say that my samples are significantly different (or not) from each other just by looking a box plots? If yes, what do I have to look at and what is the theory behind it?

I read something about notches which can be drawn at each side of the boxes, and if they do not overlap, the medians are significantly different at the 5%, but I don't know how to do this in R?

The sample size is K:19, R:35 and N:30 but I have also data that contains only 5 data points in K, 7 in R and 10 in N

Thanks a lot for your help!

• Do you have any way of knowing how many data are represented by each box? Feb 8, 2015 at 20:54
• The case where there are no points outside (Q1-1.5IQR,Q3+1.5IQR), so that you're working ony with he five number summary, is discussed here. Feb 8, 2015 at 22:17
• Are you only interested in comparing pairwise or do you want one overall hypothesis test? Feb 8, 2015 at 23:29
• Just noticed this in comments:"I am not allowed to perform any test". ... Oh, well, then my answer (and the other one) is useless. You should not talk about "significance" in your question, because that implies you want a significance test. Please fix your question so that it doesn't ask about significance (and put the part about not being allowed to test in the question itself), add all the additional information from your comments, make it clear if you're doing coursework (& if so, add the self-study tag and read the tag wiki). Feb 8, 2015 at 23:51
• Here is an example on how to add notches to your boxes in R. Feb 10, 2015 at 16:47

(This section addresses the original question)

If we were looking for some relatively formal test, then speaking in general, if there's plenty of points outside the whisker ends, you could maybe get somewhere with a generalization of a two-sample Anderson-Darling type statistic, like so. Since the Anderson Darling approach focuses more on the tails than say a Kolmogorov-Smirnov, the differences in the tails might be sufficient.

However, I think in this case (since it now appears that you know $n$'s, not just lower bounds based on the tails) that you could perhaps also construct envelopes that put lower bounds on the difference in CDFs for a Kolmogorov-Smirnov type test. This could be generalized to a k-sample statistic.

This test would have low power typically, but when you lose most of the information in your data, that's how it goes.

Outside of formal testing:

In the case of direct comparison of boxes, Arnold et al (2011)[1] give a number of rules of thumb, some of which are both simple to apply and with reasonable properties (see p5 for a list of increasingly sophisticated rules). In many stats packages, notched boxplots are available and can be used.

[1]: Arnold P., Pfannkuch M., Wild CJ, Regan M, and Budgett S (2011),
"Enhancing Students' Inferential Reasoning: From Hands-On To 'Movies',"
Journal of Statistics Education, 19:2