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• When stratification is employed to reduce the variance of an estimate of the population mean, and the population does not possess an a priori partitioning, the degree of reduction provided is affected by the method of allocation, by the number of strata, and by the choice of the stratum boundaries. However, the most effective use of either the method of allocation or the choice of the number of strata is attained when one employs the points of optimum stratification. It was shown by Dalenius (1950), when the stratification and study variables are identical, and by Dalenius and Gurney (1951), when the stratification is on an auxiliary variable related to the study variable by a known regression function, that the optimum stratification boundaries were theoretically obtainable as the solution of a system of non-linear equations. In practice, when nothing is known about the relationship between the study and auxiliary variables, Dalenius's original equations are used with the auxiliary variable as an approximation to the points of optimum stratification. Over the range of variable populations and regression functions commonly found in practice, an empirical study is made of the effect this approximation has on the reduction in variance compared with the use of simple random sampling. The effect these approximate boundary points have on the choice of the method of allocation is also considered. When something is known about the character of the regression function between the two variables, the cumbersome nature of the Dalenius- Gurney equations in practice suggests determining the stratum boundaries by employing one of a series of approximations proposed in the literature. An empirical study is made of the performance of these approximations, comparing the approximate boundaries with the optimum stratification boundaries. Finally, once the method of allocation and the mechanism for computing the stratum boundaries are chosen, the number of strata to use must be determined. An algorithm is introduced that quantifies the effect the increase in the number of strata has on the reduction in variance. With this algorithm, an emirical study is made of the approximations that have appeared in the literature designed to model the relationship between the number of strata and the precision of the estimate of the population mean.
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