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To find the change point of a normal process monitored by an X bar control chart, the first and most important step is to find a good estimate of the change point. Different from all the previous studies in which the MLE of the change point is computed from a conditional likelihood function given the signal time of the X bar control chart, in this paper we also derive the MLE of the change point from the complete likelihood function. Then, the two MLEs are compared through a series of simulations based on two criteria, MSE and a new proposed criterion. The new criterion is just the average number of points in time, ordered according to their likelihood, at which the process should be examined to find the change point. The results show that our estimator is superior to the usual MLE which was used in previous studies and this superiority is shown better by our new criterion, which is more rational, when a control chart is used.