File:Sech squared a arctanh.png

This is a plot comparing a histogram of real data with a normal distribution, a logistic distribution, and a distribution defined as

${\displaystyle f(x;\mu ,s)=\frac{a}{4s{\cosh }^2\left[a{\sinh }^{-1}\left(\frac{x-\mu }{2s}\right)\right]\sqrt{\frac{(x-\mu {)}^2}{4s^2}+1}}}$

...where ${\displaystyle a=1.6,s\approx \sigma /\sqrt{2.25}=2\sigma /3}$.

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