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I have histograms/distributions like the following that I would like to characterize/summarize by one or maybe two number(s). A mean +/- std would be one such example, but is there something better that can say something about the shape of the distribution? Most of the times the most frequent values are near the two ends, so the distributions are not normal. Is there a statistic for being the opposite of normal distribution?

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    $\begingroup$ Besides mean and standard deviation, there are summary measures like skewness and kurtosis to quantify the shape of a distribution. But that makes four numbers... $\endgroup$
    – Michael M
    Jan 13, 2018 at 8:36
  • $\begingroup$ Thanks, but for the distributions that I have, I'm not sure if skewness and kurtosis will make much sense, since they are not normal or 'normal-like' distributions. $\endgroup$
    – user191389
    Jan 13, 2018 at 15:47
  • $\begingroup$ Say they are bimodal perhaps and then give the modes? $\endgroup$
    – mdewey
    Jan 13, 2018 at 16:07
  • $\begingroup$ Thanks, that's a good suggestion, but they are not always bimodal. $\endgroup$
    – user191389
    Jan 13, 2018 at 17:01

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Could it be that your data follows the family of U-shaped distributions? https://en.wikipedia.org/wiki/U-quadratic_distribution

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  • $\begingroup$ Thanks a lot! I think this might help. I didn't know how to name a distribution like this. Are you familiar of any statistical tests that allow you to find whether a distribution is U-quadratic? Any Python/R/Matlab implementations that you know of might also be helpful. $\endgroup$
    – user191389
    Jan 13, 2018 at 15:49
  • $\begingroup$ If you are familiar with R, you could use the "fitDistr" function of my 'propagate' package which fits 31 (!) different distributions and selects the one with lowest BIC. There is a U-shaped one inside: Arcsine dist. $\endgroup$
    – anspiess
    Jan 13, 2018 at 20:26
  • $\begingroup$ Awesome! That is very helpful. Thank you! Yes, I think arcsine would fit the data best. $\endgroup$
    – user191389
    Jan 13, 2018 at 20:49
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The simple answer is: You can't.

Slightly more complex answer: The reason you can summarize a Normal distribution with two numbers is that, once you have the mean and the SD of a Normal distribution, a lot of the rest is determined. While there can be variations in any particular sample, the skewness of a Normal is 0, the excess kurtosis is 0, it's symmetric and you can estimate the number that will be in any bin of a histogram.

If you are willing to assume another distribution, then you may be able to summarize the data with one, two, three or four values, depending on the particular distribution. For instance, a Poisson distribution can be summarized with one value.

However, if you are not willing to make that assumption, then any summary will lose a lot of information. You could use range or interquartile range for spread, and median or another trimmed mean for location, but this will be much less informative than mean and sd for a Normal.

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  • $\begingroup$ Thank you for your answer. Someone mentioned about a U-quadratic distribution. Is there a way to summarize a distribution like that? Or test whether a distribution follows that pattern? $\endgroup$
    – user191389
    Jan 13, 2018 at 15:52
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What if I convert the distribution to a Cumulative distribution function like this? Can you use the beta parameters or something like that?

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