The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant do determine the dimension of a maximal torus in the group algebra.
I have investigated certain group classes as extra-special groups, abelian groups, direct products, quaternion groups, dihidral groups, semidih, groups, $S_4$ and $A_4$, $SL(2,q)$, $GL(2,q)$, meta-zyclic groups, met-abelian-groups, central products, frobenius groups and determine this sum. Also symmetric groups are no known (thanks to the answer of Alex).
My questions is whether there is a nice sum-formula (maybe with recursion) for groups like: alternating groups, p-groups (maybe special classes), simple groups, general linear, special linear groups, groups with united factor group, groups with normal p-Sylow-subgroup and abelian Hall-Complement ... as well.
In the literature there is a nice inequation saying that this sum is at least the double of the linear characters for a non-soluble groups (using the classification atlas). Maybe there are some more upper and lower bounds to derive here.