@AndreSilva I wanted to explain a bit further why I rejected the 'kurtosis' tag wiki (though I expect others will accept it in any case). Defining 4th moment kurtosis to be a measure of peakedness - as many sources do - is quite wrong (and equally, defining it as a measure of heavy-tailedness is equally wrong), as Kendall and Stuart show by a series of counterexamples.

I was reticent to reject it, but I couldn't let further perpetuation of this misunderstanding go without at least some discussion.

It would be reasonable to say kurtosis is associated with both peakedness and heavy-tailedness. If the tag wiki and extract said something like that, I think that would be good.

To clarify; one of the things Kendall and Stuart show is that you can have distributions that are more peaked than others but for which 4th moment kurtosis goes down, and they also show you can have distributions for which the tail is heavier but for which the kurtosis goes down - showing it's not either one alone. It's algebraically possible to expand the fourth moment in such a way as to show that it can be regarded as a measure of variability about mu +/- sigma.

I'll try to locate a reference for that last comment.

I think - from the name kurtosis itself (from the Greek work for arch) - it may have been intended as a measure of peakedness, but the 4th moment version can't really be said to measure that.

Here we go - Kevin P. Balanda and H.L. MacGillivray. "Kurtosis: A Critical Review". The American Statistician 42:2 [May 1988], pp 111–11

@Glen_b, thank you so much for this feedback. I am worried, because the wiki was approved and maybe, now I need to work to improve it.

I am trying to enjoy this <2k reputation period, because all my edits are peer reviewed, so receiving these feedbacks puts me on the right track.

Do you have the full reference of Kendall and Stuart, so I can study more about kurtosis?

It's their 3 volume "Advanced Theory of Statistics" (originally Yule&Kendall, then Kendall&Stuart then Kendall, Stuart and Ord, then Stuart & Ord , ...). I don't know for certain which volume, but I assume Volume I. It's 25 years since I saw that part and probably 20 years since I have even held a copy at all.

Ok. I think Nick Cox mentioned that either "many authors describe kurtosis as measuring tail weight"

A good starting point is the Balanda and MacGillivray paper.

I don't think anybody can be blamed for thinking it really measures peakedness since so many sources assert that. Similarly, I don't think I could blame anyone for thinking it really measure heaviness of tail, since so many other sources assert that. It's really a bit of each.

There are a number of other sources on the issue but those two (B&M and K&S) are probably more than sufficient.

I will study more kurtosis during this week, and I will try to give another suggestion to improve the wiki (if someone did not that first). It really seems to be a polemic issue. Thanks for those considerations.

Thanks again for this clarification. Have a nice week!

@AndreSilva Looking around, it looks like chapter 3 of the 1967 edition (which is definitely Vol. 1) discusses kurtosis, so it might be in there. What's in the book varies from edition to edition.

If anyone is interested, I would like to share two answers I received and that helped me to change the background color of accepted answers on CV SE and GIS SE: this one, and this one.