I have two datasets. Each contains integer data points, ranging from 0 to 34 and they follow an approximately half-normal distribution, as 0 occurs very frequently and 34 occurs only once.

I would like to test whether one dataset shows significantly lower values than the other. I have never worked with half-normal distributions before, so my questions are:

  1. Should I first test whether I am allowed to consider this a half-normal distribution? If so, how?

  2. What test do I use to assess whether means or medians are different? T-test? Mann-Whitney U test? Other? To decide this, does it matter that my data points are discrete/integer and thus not continuous?

EDIT: Here is a boxplot summarizing the data. "+" shows the mean. All individual data points are shown from left to right per set. Total number of data points on top. It looks pretty clear to me that there is a difference, but I also have some less clear cases and I am not sure which statistical test is formally correct here.

Parallel boxplots comparing the two groups

  • $\begingroup$ Could you post some plots of your data? $\endgroup$ Mar 5, 2019 at 10:15
  • $\begingroup$ Please see edit above, @kjetil-b-halvorsen. Thanks. $\endgroup$ Mar 5, 2019 at 12:04
  • $\begingroup$ From the plots it is very clear that at least the medians are different. And variances cannot be equal! $\endgroup$ Mar 6, 2019 at 20:21
  • $\begingroup$ Why do you need a statistical test for these data, which are clearly different (or are the datapoints dependent and is the sample size effectively smaller)? What do you want to show with the statistical test? You have doubts that these two sets could be samples from the same distribution? $\endgroup$ Mar 19, 2022 at 20:40

1 Answer 1


I would like to test whether one dataset shows significantly lower values than the other.

You will need to formalize what that means. Different means? Different medians? Or that $P(X \le Y) \not= \frac12$? Or something else. I would bystep the question about which theoretical distribution "the data follows", with your sample size bootstrapping should be adequate. Or if you instead just want to test the null hypothesis that the distributions are equal, you could use a permutation test.

If you could post (a link to) your data I could make some examples.

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    $\begingroup$ I wonder whether the means or the medians are significantly different. I say OR, because I do not know which I should test. On the one hand, the data are not normally distributed (as per D'Agostino & Pearson normality test), suggesting (to the best of my knowledge) using a (non-parametric) Mann-Whitney U test to see whether medians are significantly different. On the other hand, the data may follow a half-normal distribution (but I am not sure, so you are probably right to stay away from this question), but this could mean that I need to test whether the means are statistically different. $\endgroup$ Mar 5, 2019 at 13:02
  • $\begingroup$ I am not familiar with bootstrapping or permutation tests, but I'll look into that. Thanks for the suggestions. Sorry, I am not able to post the data, but I appreciate your help. $\endgroup$ Mar 5, 2019 at 13:03
  • $\begingroup$ What does the measurements/counts represent? What do you use them for? what is the context? That should help you decide which descriptive measures are relevant for you. $\endgroup$ Mar 5, 2019 at 13:04
  • $\begingroup$ It's a biological question, it's complicated, but is that important (with all due respect)? Isn't this a purely statistical question? Thank you very much. $\endgroup$ Mar 5, 2019 at 13:08
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    $\begingroup$ Yes, it is important. Statistics is a way of helping YOU answer questions of interest to YOU. But statistics cannot tell YOU what YOU should be interested in knowing! So, deciding which descriptive measures are relevant for YOU is a substantial matter, not a statistics matter (although some stat knowledge can help.) When that is decided, statistics enter. $\endgroup$ Mar 5, 2019 at 13:15

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