"fit" and "cftool" fold the "2" term in the exponent denominator into the estimate for "c1". That is, the "c1" estimate fitted by "fit" will be greater than the "c1" estimate fitted using the Wikipedia formulation by a factor of sqrt(2).
To see this, we can generate standard normal data and then fit a Gaussian model:
pd = makedist('Normal');
x = -5:.1:5;
y = pdf(pd,x);
f = fit(x',y','gauss1')
pd =
NormalDistribution
Normal distribution
mu = 0
sigma = 1
f =
General model Gauss1:
f(x) = a1*exp(-((x-b1)/c1)^2)
Coefficients (with 95% confidence bounds):
a1 = 0.3989 (0.3989, 0.3989)
b1 = -9.1e-13 (-1.351e-07, 1.351e-07)
c1 = 1.414 (1.414, 1.414)
The "c1" estimate is approximately the square root of 2, as expected.
