The classical method for the allocation of tolerances permits component variables to be anywhere within their tolerances and ensures that when the components are assembled, the resultant variable will be within its given specification. In general, only small proportions of the component variables are near their limits and therefore the variability of the resultant variable is less than that allowed by the classical method. Probability theory, by estimating this reduced variability, allows

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... sed tolerances for the component variables and a consequent reduction in their cost of manufacture. A review of previous work on the application of probability theory to the allocation of tolerances shows that previous authors have assumed that component variables have their expectations exactly equal to their design values and that their distributions are unimodal and symmetrical. An investigation of actual distributions indicates that neither of these assumptions is fulfilled, since the means of batches of components generally differ from their design values and bimodal and skew distributions occur. To allow for these types of variation found, two Models are proposed for specifying the dispersion of component variables about their design values. Unlike previous methods, these specifications allow objective -11inspection and control by conventional quality control methods. Limits for the dispersion of resultant variables which are linear functions of component variables conforming to the Models are provided in the form of probability inequalities and graphs. A method is given to transform non-linear functions between component and resultant variables into linear functions, if this is possible. A systematic procedure for the probabilistic allocation of tolerances to component variables is presented. It is optimum from an economic point of view and can allow for the non-independence of some of the component variables. Two examples demonstrate the procedure and a further example is used to provide an empirical verification of the procedure.