In this paper, following a very recent and new approach of Aizpuru et al. (Quaest. Math. 37:525-530, 2014), we further generalize a concept of α-density to that of f α $f_{\alpha}$ -density, where f is an unbounded modulus and 0 < α ≤ 1 $0 < \alpha\leq1$ . As a consequence, we obtain a new nonmatrix convergence method, namely f-statistical convergence of order α or S α f $S_{\alpha}^{f}$ -convergence, which is intermediate between the ordinary convergence and the statistical convergence of order α. We also introduce a new concept of strong Cesaro summability of order α with respect to a modulus function f, and finally we investigate the relationship between the set S α f $S_{\alpha}^{f}$ of all f-statistically convergent sequences of order α and the set w α f $w_{\alpha}^{f}$ of all strongly Cesaro summable sequences of order α with respect to f.