Mixing up Standard Errors and Standard Deviations

Over at the Non Significance blog, the author describes the case of a paper that has some strange descriptive statistics:

What surprised me were the tiny standard deviations for some of the Variable 1 and 2, especially in combination with the range given.

In the blog post, the author outlines his approach to make sense from the descriptive values. It seems to be likely that the reported Standard Deviations (SD) are actually Standard Errors of the Mean (SEM). I’d like to add to this blog post one argument based on calculus and one argument based on simple simulations to show that SEM’s are indeed much more likely than SD’s.1