Claiming it should be a "normal distribution" (aka a bell curve) is a major assumption based on there is a "correct" temperature that remains fixed over time.

Claiming it should be a "normal distribution" (aka a bell curve) is a major assumption based on there is a "correct" temperature that remains fixed over time. To the best of my knowledge, few places have tracked temperature longer than the last 150 years and beyond that we have estimates and calculations that have variances/tolerances built in. Is 150 data points for a given location enough to KNOW both the "correct temperature" and that it does/doesn't move?