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Some have stated that kurtosis is the "movement of probability mass from the “shoulders” of a distribution into its center and tails" where "center" is defined as the range between $\mu \pm\sigma$. I was trying to find a proof for this but was unsuccessful. Can anyone point me to a proof of whether (i) larger kurtosis implies greater probability in the $\mu \pm\sigma$ range, or (ii) greater probability in the $\mu \pm\sigma$ range implies larger kurtosis?

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    $\begingroup$ Given that you already know the answer to this, I think a more broadly beneficial question would perhaps ask whether anything can be said about more in the tails or more near the peak when kurtosis increases. $\endgroup$ – Glen_b -Reinstate Monica Nov 25 '17 at 9:48
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    $\begingroup$ If my internet connection stays up, I should have time to get back to this over the next couple of days. Which definition of more in the tails is that? $\endgroup$ – Glen_b -Reinstate Monica Dec 7 '17 at 1:14
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    $\begingroup$ The 3 theorems are (i) kurtosis is between E(Z*4*I(|Z|>1) and E(Z^4*I(|Z|>1) +1, for all distributions with finite 4th moment, (ii) when the distributions are restricted to continuous with decreasing density of Z^2 on (0,1), then the +1 can be replaced with +.5, and (iii) for any sequence of distributions having kurtosis tending to infinity, E(Z^4*I(|Z|>b))/kurtosis -> 1, for all real b. The definitions of tailweight implicit in these 3 theorems are, for (i) and (ii), E(Z*4*I(|Z|>1), and for (iii), E(Z*4*I(|Z|>b). $\endgroup$ – Peter Westfall Dec 12 '17 at 1:14
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I saw a very good reference that pointed out that these intuitive/graphic ideas are not really sound. Kurtosis is often explained as (1) more mass in the tails; (2) less at the center; and (3) some shift relative to the 'shoulders'. I have also seen the claim, as you say, where 'it shifts from the shoulders both in and out, so there is more in both the shoulders and tails.' But all of that is very geometrical, and it raise the question of 'more than what'? And it presumes 'shoulders', which are a vague concept.

I will try to dig up the reference, but I don't think you can prove what you claim. Imagine taking a normal distribution, then take all points within 1-$\sigma$ and put them out at >3-$\sigma$. That will have more kurtosis, I believe, but zero mass in the center.

I will try to dig up that reference, since he did a good job of giving both examples and pointing out that kurtosis is what it is - and while it may have some relation to variance or SD, it needs to be understood (like all moments) for what it is mathematically (which is simply a statistic based on expectations of points raised to a power).

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