What is the reason behind the seemingly arbitrary factor of $\displaystyle\frac{1}{{2}}$ in the exponent of the Gaussian function?
Sometimes, as in this Wikipedia article on the Gaussian integral, the factor $\displaystyle\frac{1}{{2}}$ is omitted and it's not obvious why one should include it at all. It's just another factor one has to take care of (i.e. a possible source of error when doing calculations by hand).
As far as I can tell, it just scales the width parameter (e.g. the standard deviation $\displaystyle\sigma$ or the full width at half-maximum) by a "strange" factor of $\displaystyle\sqrt{{{2}}}$ and seems to serve no real purpose.