rank of sum of random elements of Pauli group

Question

The Pauli group for n-qubits is defined as $G_n=\{I,X,Y,Z\}⊗n$, that is as the group containing all the possible tensor products between $n$ Pauli matrices. I take the real version where $X=[[0,1],[1,0]]$,$Z=[[1,0],[0,-1]]$,$Y=XZ$. For any element $S_i$ of this group, $S_i^2=\pm I$. Now I take the sum of $m$ random elements of this group and calculate its rank. For $m$ even, there seems to be a definite distribution with full rank being the most probable. For $m$ odd, it seems that the sum is always full rank! This is surprising. Can anyone provide a theoretical explanation for these two cases. I think this might also generalize for random involutions.