Many believe that severe intellectual impairment, blindness or dying young amount to serious harm and disadvantage. It is also increasingly denied that it matters, from a moral point of view, whether something is biologically normal to humans. We show that these two claims are in serious tension. It is hard explain how, if we do not ascribe some deep moral significance to human nature or biological normality, we could distinguish severe intellectual impairment or blindness from the vast list of seemingly innocent ways in which we fail to have as much wellbeing as we could, such not having super-intelligence, or not living to 130. We consider a range of attempts to draw this intuitive normative distinction without appealing to normality. These, we argue, all fail. But this doesn't mean that we cannot draw this distinction or that we must, implausibly, conclude that biological normality does possess an inherent moral importance. We argue that, despite appearances, it is not biological normality but rather statistical normality that, although lacking any intrinsic moral significance, nevertheless makes an important moral difference in ways that explain and largely justify the intuitive distinction.