What number of points is required to fit Gaussian Distributions?

A Gaussian distribution has an exponential term and 2 variables. Its log will turn it linear. But I am not able to figure out how many points will suffice to fit a gaussian distribution and why? It is recommended to be 5 to 12 points in some articles but no concrete reasons are given.

Curve Fitting

Regression Analysis

Gaussian Processes

Spectroscopy

Statistics

Samuel Arba Mosquera

SCHWIND eye-tech-solutions GmbH

In some other papers you can find as a general rule a minimum of 20 observations per each degree of freedom (sometimes DoF + 1), this will make 60 (or 80) observations for a robust fit. Why 20, is not clear to me, I will check my secondary references.