Construction of Strata Boundaries in Audit Sampling
The cumulative square root of the frequency method1 is a generally accepted statistical sampling method to construct strata boundaries. It has been widely used for decades in audits. It continues to be among the most prevalent and recommended design methods for consideration in revenue sampling manuals, audit literature, statistical sampling research, and software.
Even with this historical use and the abundance of statistical literature on design methods, there is little guidance on determining the appropriate width of the interval or, stated differently, the number of class intervals. This is partly because there is no existing theory2 that determines the best interval width.
Much of the statistical sampling literature arbitrarily recommends many classes, or in other words, narrow interval widths (Cochran 1961; Cochran 1977; Roberts 1978; Hedlin 2000; Hogan 2010). Notwithstanding the interval width chosen, the statistical literature suggest that varying interval widths is negligible.
Ryan statisticians recently explored the effects of varying interval widths and presented a paper on the conclusions at the 2019 Joint Statistical Meeting of the American Statistical Association (ASA). The research did not seek a theoretical solution for determining optimal internal widths but puts forth a data-driven analysis exploring width effect on numerous simulated populations with skewed distributions, as is commonly found in accounting populations.
The research3 concluded that interval width has a measurable and meaningful effect on the cumulative square root of the frequency method and the associated accuracy and precision of the estimate. With this new research, the guidance of “many classes” or “narrow interval widths” should not be viewed as a “one size fits all” solution. The cumulative square root of the frequency method should be used dynamically toward an understanding of a population’s unique attributes and consciously to create desirable statistical qualities of accuracy and precision. The research is important to companies that undergo or conduct audits or reviews with a sampling component and highlights the need for trained sampling practitioners who knowledgeably evaluate the statistical and data qualities of a population to manage all aspects of sampling risk.
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1 Dalenius, T. and Hodges, J., “Minimum variance stratification,” Journal of the American Statistical Association (1959), 88–101.
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