  In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ for $n\ge
6$ with reversibility $\lambda$ and flag curvature $K$ satisfying
$(\frac{\lambda}{\lambda+1})^2<K\le 1$, either there exist infinitely many
prime closed geodesics or there exist $[\frac{n}{2}]-2$ closed geodesics
possessing irrational average indices. If in addition the metric is bumpy, then
there exist $n-3$ closed geodesics possessing irrational average indices
provided the number of closed geodesics is finite.
