The concept of deferred Nörlund equi-statistical convergence has recently been studied by Srivastaval et al. (see,[19]). In this paper, we have introduced the notion of equi-statistical convergence, statistical point-wise convergence and statistical uniform convergence in conjunction with the deferred statistical convergence and established a inclusion relation between them. Moreover, we have applied our idea (presumably new) of the deferred equi-statistical convergence to prove a Korovkin-type approximation theorem and demonstrated that our theorem is a non-trivial extension of some well-established Korovkintype approximation theorems which ware proved by some earlier authors. Furthermore, we have established the rate of deferred equistatistical convergence and accordingly proved a theorem. Finally, some concluding remarks and fascinating cases are shown here in support of our definitions and results.