In Wikipedia's article on Cohen–Macaulay rings, the following geometric version of miracle flatness is stated, see this link:
Let $X$ be a connected affine scheme of finite type over a field $K$ (for example, an affine variety). Let $n$ be the dimension of $X$. By Noether normalization, there is a finite morphism $f$ from $X$ to affine space $\mathbb A^n$ over $K$. Then $X$ is Cohen–Macaulay if and only if all fibers of $f$ have the same degree.
Exercise 18.17 of Eisenbud's commutative algebra book is cited, but this exercise actually has more assumptions and says something that, for me, sounds completely different. Is the above statement even true? Or is this a typo and there should be "dimension" instead of "degree"?