In this paper, we prove that on every Finsler n-sphere (S n, F) for n ≥ 6 with reversibility λ and flag curvature K satisfying (λλ+1)2<K≤1 , either there exist infinitely many prime closed geodesics or there exist [n2]−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 prime closed geodesics is finite.