# Intuition Behind Kurtosis of Binomial Distribution

I'm trying to get a good understanding of the higher moments of the Binomial Distribution; it's an important building block for more complex distributions so I want to get a strong intuition for this.

The thing that confused me was that lower p has higher kurtosis.

Looking at PMFs, it is clear the wing distributions are more "peaked" but the values don't seem to have a "fatter tail". If the kurtosis is the fourth power of the differences vs the mean, I would naively think the Binomial Dist with p=.5 has higher kurtosis; it seems more spread out! I would have thought that the more extreme values of p are, the tighter values would be clustered around that. Because of this, the plot of variance makes sense to me. So it makes sense to me that p=.5 has highest variance and least skew, but why would it also have the least kurtosis/thinnest tails? The issue here is scaling by the standard deviation. The tail points are not any further out in absolute terms when $$p$$ is extreme, but they are further out as a multiple of the standard deviation. The standard deviation is $$\sqrt{p(1-p)}$$, so being nearly 1 unit away is further away in standard deviations as $$p$$ gets extreme.
The scaling for the kurtosis is the fourth power of the standard deviation (so $$(p(1-p))^2$$, so the kurtosis goes off to infinity as $$p$$ goes to zero or one, even though the raw fourth moment doesn't.