Suppose \((M,g)\) denotes a \(d\)-dimensional Riemannian manifold and \(p\in M\). The mean sectional curvature of a vector \(v\in T_pM\) is determined by the Ricci curvature. To be more precise,
\[ m(v):= \frac{1}{\Vol (\mathbb{S}^{d-2})} \int_{\mathbb{S}^{d-2}} \sec (v,w) \,d\sigma(w) = \frac{1}{(d-1)} \Rc(v,v). \]This follows by applying the sectional curvature formula.