In this paper, we further generalize recently introduced summability methods in [23] (where ideals of $\mathbb N$ were used to extend certain important summability methods) and introduce new notions, namely, $\mathcal I$-statistical convergence of order $\alpha$, where $0<\alpha<1$ by taking nonnegative real-valued Lebesque measurable function in the interval $(1\infty)$. We mainly investigate their relationship and also make some observations about these classes. The study leaves a lot of interesting open problems.