The real issue with stereotypes isn't that they are false but that people tend to apply them in situations where they are inaccurate because of conditionization on other information.

Whether or not it's true that on average blacks tend to be less conscientious workers or asians more conscientious [1] isn't really that important when making a hiring deciscion. What you'd want to know is whether -- conditional on the information you have -- that variable still has the supposed relationship. People aren't good with conditionalization so they tend to make the assumption that the relationship continues to hold but it's quite plausible that the relationship is actually reversed once you know information like what college someone went to. And that's really what makes stereotypes so pernicious -- they aren't dispelled even with evidence that should suffice.

And it's such a tempting mistake to make even mathematics grad students can't help but feel the pull [2]. That's what makes stereotypes dangerous regardless of whether they are true in some context.

--

1: I'd be quite surprised if conscientiousness wasn't usually less for people raised in low income situations generally so it's plausible if -- controlling for nothing else -- this shouldn't be expected merely from average income statistics.

2: Even math graduate students who intellectually know better can't help but emotionally feel that if women are on average worse at math it calls into question the abilities of the women in a graduate math program. Yet, of course, what's true on average for women tells you nothing about what's true for women conditional on scoring well in math classes/tests/getting accepted into a given program.

I don't bring this up to claim it's true or false merely to note how difficult it is to emotionally grapple with conditionalization.