  Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is the
power of an odd prime, and let $\mathrm{GSp}(2n, \mathbb{F}_q)$ and
$\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$ denote the symplectic and orthogonal
groups of similitudes over $\mathbb{F}_q$, respectively. We prove that every
real-valued irreducible character of $\mathrm{GSp}(2n, \mathbb{F}_q)$ or
$\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$ is the character of a real
representation, and we find the sum of the dimensions of the real
representations of each of these groups. We also show that if $\boldsymbol{G}$
is a classical connected group defined over $\mathbb{F}_q$ with connected
center, with dimension $d$ and rank $r$, then the sum of the degrees of the
irreducible characters of $\boldsymbol{G}(\mathbb{F}_q)$ is bounded above by
$(q+1)^{(d+r)/2}$. Finally, we show that if $\boldsymbol{G}$ is any connected
reductive group defined over $\mathbb{F}_q$, for any $q$, the sum of the
degrees of the irreducible characters of $\boldsymbol{G}(\FF_q)$ is bounded
below by $q^{(d-r)/2}(q-1)^r$. We conjecture that this sum can always be
bounded above by $q^{(d-r)/2}(q+1)^r$.
