  We investigate the effect of the average scalar curvature on the conjugate
radius, average area of the geodesic spheres, average volume of the metric
balls and the total volume of a closed Riemannian manifold $N$ (or more
generally $N$ with finite volume whose negative Ricci curvature integral on
$SN$ is finite). For example, we prove that if the average scalar curvature is
larger than the lower bound of the normalized Ricci curvature, then we can
improve the Bishop-Gromov estimate on the average volume of the metric balls of
any size. We also prove the monotone decreasing property of a certain geometric
integral when the average scalar curvature has a lower bound. This leads to a
comparison theorem of the average total mean curvature of geodesic spheres of
radius up to $\mathrm{inj}(N)$.
