[MaeveChondrally (OP)]

s=c*NkT*integral from -inf to inf of    ln(p(x)*(1-p(x))) dx
c is a constant based on the normal distribution.  N is the number of moles or it could be the number of data points in the numerical distribution., k is boltzmans constant and T is the temperature in kelvin. This formula is absulutely derivable from Statistical Mechanics and came from examining equations in Statistical Mechanics, 3rd Edition by Raj Pathria.
if you just want to assess the entropy of the normal distribution then
set N=1Mol=6.023x10^(23) particles per mol and k=boltzmans constant and T=298.13Kelvin
and p(x)=1/sqrt(2pi)*exp(-x^2/2) where x=tan(theta) and dx = sec^2(theta) dtheta
and it can be discretized with h=(pi/2)/1025; so that it can be numerically integrated with the simpsons method 5th order or 2nd order as you wish but do not include the pi/2 value, everything else but.
so p(theta)=1/sqrt(2*pi)*exp(-tan(theta)^2/2)*sec(theta)^2
calculate this probability value for every point from -pi/2 to pi/2 except the two endpoints and sum it all up with simpsons method to see what s is.
Numerical distributions can be calculated with this method as well and all other distributions can be calculated with this method too.
If the entropy value is within 1% of the entropy of a different data set, then at the 1% level it is a normal distribution.  differences in mean and standard deviation should be filtered out of numerical datasets by normalizing the data before testing its entropy to see what distribution it is most like.
best regards,
Maeve Chondrally
aka Chondrally