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Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is thepower of an odd prime, and let $\mathrm{GSp}(2n, \mathbb{F}_q)$ and$\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$ denote the symplectic and orthogonalgroups of similitudes over $\mathbb{F}_q$, respectively. We prove that everyreal-valued irreducible character of $\mathrm{GSp}(2n, \mathbb{F}_q)$ or$\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$ is the character of a realrepresentation, and we find the sum of the dimensions of the realrepresentations of each of these groups. We also show that if $\boldsymbol{G}$is a classical connected group defined over $\mathbb{F}_q$ with connectedcenter, with dimension $d$ and rank $r$, then the sum of the degrees of theirreducible 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 connectedreductive group defined over $\mathbb{F}_q$, for any $q$, the sum of thedegrees of the irreducible characters of $\boldsymbol{G}(\FF_q)$ is boundedbelow by $q^{(d-r)/2}(q-1)^r$. We conjecture that this sum can always bebounded above by $q^{(d-r)/2}(q+1)^r$.

Character degree sums and real representations of finite classical groups of odd characteristic