Consider a variety $X$. As the title suggests I would like to know if the hypotheses
1) $X$ is Cohen-Macaulay
2) the singular locus of $X$ has codimension $>1$
imply that $X$ is normal.
I hope it is true or trivial. Any suggestion or reference? If it was true, are there wild conditions for 1) such that used with 2) give normality for X?