  I. Schur studied double covers $\widetilde{\Sym}^{\pm}_n$ and
$\widetilde{\Alt}_n$ of symmetric groups $\Sym_n$ and alternating groups
$\Alt_n$, respectively. Representations of these groups are closely related to
projective representations of $\Sym_n$ and $\Alt_n$; there is also a close
relationship between these groups and spinor groups. We study the essential
dimension $\ed(\widetilde{\Sym}^{\pm}_n)$ and $\ed(\widetilde{\Alt}_n)$. We
show that over a base field of characteristic $\neq 2$,
$\ed(\widetilde{\Sym}^{\pm}_n)$ and $\ed(\widetilde{\Alt}_n)$ grow
exponentially with $n$, similar to $\ed(\Spin_n)$. On the other case, in
characteristic $2$, they grow sublinearly, similar to $\ed(\Sym_n)$ and
$\ed(\Alt_n)$. We give an application of our result in good characteristic to
the theory of trace forms.
