File:Sech squared a arctanh.png

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

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

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