# Audit Sampling with ACL

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Audit Sampling with ACLWhen and why would you sample in audit?Basically, you can expect to sample whenever you:•need to make an estimate concerning some large set of transactions, and•cannot find a more cost-effective way to arrive at the number.If the data you are using are 100% digital, then typically you can review the entire database using ACL commands in less time than it would take to pull a sample from manual files. Instead of testing 500 transactions by hand, you can get an exact answer for 5,000, 50,000, or 5 million transactions, with far less effort.However, if the data are only available in paper files, or if they require a great deal of processing for each item, such as generating confirmation letters or checking actual inventory, then sampling may be faster and cheaper.The key to this more complex approach is the idea of ‘tolerable errors’.Instead of doing a calculation that only works when no errors are found, ACL can collect some additional information before starting, then compute a sample confidence level that allows for up to a certain number of errors.So long as the tolerable number of errors is not exceeded, the sample confidence level can be treated as the population confidence level.To accomplish this, ACL asks for an estimate of the expected error rate (EER) in the population and for the upper error limit (UEL) that you will accept within your confidence level.For example, you might expect an error rate of 1% and, at the same time, want to be 95% confident that the error rate does not exceed 2%.The key item that ACL delivers using your estimate of the EER and UEL is the number of tolerable errors, the number of errors that a sample can contain without forcing a recalculation of the number of errors in the population based upon the actual number of errors in the sample.In general, the EER can be based on similar sampling projects in the past or interviews with the staff whose data you are sampling. The UEL is more difficult to choose and has a much bigger potential impact on the amount of work you must do. Consider these case parameters:population = 12,000EER = 1%sample confidence level = 95%Given these parameters, if you enter 2% for the UEL into ACL, the required sample size is given as 593 records. If you enter 1.5%, the required sample size is 1,700 records. Clearly, the UEL has a significant impact on the work involved in sampling, so to make choosing a UEL easier, ACL allows you to recalculate various values for the UEL before saving the result. This is sometimes necessary because your first estimates of the sample confidence level and the UEL may lead to an impractical sample size. By helping you choose an appropriate UEL and the sample size required to stay within that limit, ACL can alleviate the need for a recalculation by ensuring the sample and population confidence levels will be the same.ACL does not use the binomial distribution:In general, a binomial equation describing the outcome of n events (or records) will have n + 1 terms, and each term will involve multiplying probabilities to the power n. So the total computations are equal to n(n + 1).As n increases in value, the equation quickly becomes impractical because there are too many terms to compute.As the number of events grows, the number of computations required to solve the theorem increases geometrically.ACL does use the Poisson distribution:approximation to the binomial equation for very large sets that did not depend on the population size, N. Instead, it depended on the total number of expected successes, v, which is obtained by multiplying the population size, N, by the probability of success, p.It must be pointed out, however, that though Poisson is the method of choice, it does have a significant shortcoming.For small data sets, using the Poisson distribution leads to unnecessarily large sample sizes, mainly because it does not consider population size.In extreme cases, for a given sample confidence level, the Poisson distribution might indicate double the number of sample items than the binomial approach.The cost of pulling so many items for review can be substantial.Poisson may even require a sample size greater than the entire population, which is, of course, impossible.It is a fair question to ask why ACL chose Poisson as its formula for sampling given this shortcoming.The main reason is that the Poisson distribution can cover the whole range of possible values, while it becomes impossible to use the binomial distribution beyond a very limited range of cases. If you must choose one method for all statistical sampling, the Poisson distribution is the only available tool.And Poisson’s tendency toward large sample sizes can also be considered an advantage because there are good reasons to exceed minimum sample sizesThe simplest form of sampling, which is called record sampling in ACL, each transaction has an equal chance of being chosen.If there’re many low-value transactions, the resulting sample has a disproportionate number of small items, and you wind up with not enough useful information about the high-value items.The solution is the monetary unit sampling (MUS), sometimes called probability proportional to size (PPS) sampling. The MUS method counts each dollar of each transaction as a sample unit. It ensures that items of high value are given greater weight than items of low value because the high-value items have more dollar-units.Document Outline

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