Let \(B\) be the unit ball in \(\mathbb{R}^n\) and \({\mathcal H}^p(B)\) be the usual harmonic Hardy space on \(B\), where \(1< p<+\infty\). The author characterizes those harmonic functions \(u\) on \(B\) that belong to \({\mathcal H}^p(B)\) in terms of the convergence of the integral \[ \int_B|u(x)|^{p-2}|\nabla u(x)^2(1-|x|^2) dx, \] and also establishes connections between membership of \({\mathcal H}^p(B)\) and the growth of the spherical means of \(|\nabla u|^q\) for certain values of \(q\). The final section of the paper presents related results for Bergman spaces of harmonic functions on \(B\).