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The setting of limits on X̄ control charts, as well as other techniques employed in industrial statistics, are based on an assumption of normality justified by the central limit theorem. The theorem essentially states that, under general conditions, the distribution of sample means will approach normality for large sample sizes. This paper addresses itself to the question, “How large?” and “To what approximation?”. By numerically inverting the appropriate characteristic functions, tables are provided which show the manner of approach to normality for various underlying distributions and sample sizes. Also, sample sizes are given such that, at selected points, the sum of the tail areas of the distribution of sample means will be within given values. Applications to control charts for the mean are discussed in relation to the results obtained by Shewhart.
Schilling et al. (1976) studied this question.