I’d say it’s related to the central limit theorem, but would be cautious about equating the two. We would probably expect a Gaussian distribution from a variable which is the sum or product of a lot of component parts (i.e. lots of different estimator methods), but we wouldn’t necessarily expect the mean to coincide with the true value unless some of those estimator methods were reliable, and they didn’t collectively skew the distribution in one direction.
(and nit-picking, it’s “a well-defined population mean and population standard deviation”, which is required for defining the distribution. If you can’t trust your sample mean and sample SD to approximate your population mean and SD, it’s no longer reliable, and you’d have to use something else, like a t-distribution)
I’d say it’s related to the central limit theorem, but would be cautious about equating the two. We would probably expect a Gaussian distribution from a variable which is the sum or product of a lot of component parts (i.e. lots of different estimator methods), but we wouldn’t necessarily expect the mean to coincide with the true value unless some of those estimator methods were reliable, and they didn’t collectively skew the distribution in one direction.
(and nit-picking, it’s “a well-defined population mean and population standard deviation”, which is required for defining the distribution. If you can’t trust your sample mean and sample SD to approximate your population mean and SD, it’s no longer reliable, and you’d have to use something else, like a t-distribution)