It's correct that a normal distribution is assumed in physics, but you are not correct that a standard deviation is defined in terms of a normal distribution.
The present conversation got started by someone non-technical asking what 5 sigma meant in a physics context. Within that context, and with the intent to avoid exotic digressions, my reply is correct.
In physics, a sigma value maps to a p-value, and that relationship is most often defined with respect to a normal distribution. Therefore, in most cases, to go from a sigma value to a p-value, one performs this integral:
Specifically, the above definite integral, when performed with arguments of 5 and +oo, yields the often-quoted one-tailed p-value for "5 sigma", which we can get here as well:
It seems Wolfram Alpha makes the same default assumption I do: a normal distribution. If I weren't answering an inquiry from someone who wanted the clearest possible answer, I might have replied differently.
