Not every error bar represents a Gaussian, if for no other reason that most error bars aren’t symmetric.
The error bars for small sample size relative to population size are Gaussian.
Error due to a non-representative sample can have a variety of shapes, but their distribution might also be unknown. We do frequently, almost implicitly, assume unknown distributions to be Gaussian, but we should recognize that’s not necessarily a true fact about the universe.
Not every error bar represents a Gaussian, if for no other reason that most error bars aren’t symmetric.
The error bars for small sample size relative to population size are Gaussian.
Error due to a non-representative sample can have a variety of shapes, but their distribution might also be unknown. We do frequently, almost implicitly, assume unknown distributions to be Gaussian, but we should recognize that’s not necessarily a true fact about the universe.