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P erformance measures of control charts with estimated parameters are random variables and vary significantly across reference samples. In this context, a recent idea has been to study the distribution of the realized (or conditional) in-control average run length (CARL 0 ) [or, equivalently, the conditional false-alarm rate (CFAR)] for a set of estimates from a given reference sample and apply the exceedance probability criterion (EPC) to design control charts that ensure a desirable in-control performance. Under the EPC, the probability that the CARL 0 (or the CFAR) is at least (or at most) equal to a specified value is guaranteed with a high probability, which helps prevent low in-control ARL's (or high false-alarm rates) from occurring. In order to apply the EPC, the c.d.f. of the CARL 0 (or the CFAR) is necessary. For the two-sided Shewhart Xbar control chart, under normality, we derive the exact c.d.f. of the CARL 0 and the CFAR, currently not available in the literature. Using these key results, we calculate the minimum number of Phase I samples required to guarantee a desired in-control performance in terms of the EPC. Since the required amount of data can be prohibitively large, we also provide exact formulas for adjustments to the control limits for a given amount of Phase I data; some tables are provided. Our adjustment formulas give more accurate results compared to some available methods. The impact of these adjustments on the out-of-control performance of the chart is examined in detail. A summary and some recommendations are provided.

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