We introduce the Gaussian part of a compact quantum group $\QG$, namely the largest quantum subgroup of $\QG$ supporting all the Gaussian functionals of $\QG$. We prove that the Gaussian part is always contained in the Kac part, and characterise Gaussian parts of classical compact groups, duals of classical discrete groups and $q$-deformations of compact Lie groups. The notion turns out to be related to a new concept of "strong connectedness" and we exhibit several examples of both strongly connected and totally strongly disconnected compact quantum groups.