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The celebrated least mean square (LMS) algorithm is the widely used system identification approach which can be easily implemented. With the assumption of no dependence among the tapped-delay input vectors, the mean square analysis of LMS algorithm based on independence theory is only an approximate description of its convergence behavior, especially when updated with a large step-size. In this paper, we propose a modified mean square error (MSE) update formula that exactly describes the convergence process of LMS for tapped-delay independent identical distributed (i.i.d.) input data. The qualitative analysis is presented to reveal the significance and rationality of the proposed formula. Moreover, the simulations in various conditions validate that, even with a large step-size used, the study curves produced by the proposed formula are much more accurate in predicting the convergence behavior, compared with that based on independence assumption.