Statistical Applications in Accounting

This article focuses on how statistical sampling techniques are utilized in the field of accounting. Techniques such as auditing sampling are discussed. The role of the American Institute of Certified Public Accountants in the development of guidelines for this approach is highlighted. In addition, there is a historical overview of how statistical techniques were introduced into the field of accounting.

Keywords American Institute of Certified Public Accountants; Audit Sampling; Dollar Unit Sampling; Simple Random Sampling; Statement on Auditing Standard; Statistical Correlation Technique; Statistical Sampling Technique

Accounting > Statistical Applications in Accounting

Overview

Many have argued that statistics is important to the field of accounting. The tools of statistics can assist accountants with more effectively performing their job. In addition, "there is definite evidence in accounting periodicals of an increasing interest in the use of statistics, especially statistical sampling techniques, in accounting" (McGurr, 1960, p. 60). Many scholars have recognized that the two fields can yield creative results when they are combined. As a result, there was considerable interest in relating statistical methods to accounting, auditing, and management control during the post-war years (Trueblood & Cooper, 1955). Fan & Zhang (2012) show that statistical reporting is key to quality accounting practices.

Methods for Combining Statistics & Accounting

Trueblood (1953) provided some foundational principles for guiding and organizing research in this area. Based on his report, the principles were:

• Collaboration between the two disciplines is crucial to the success of the partnership between the two fields. The accountant must be willing to work closely with the statistician when stating and defining accounting problems and objectives. The statistician is responsible for understanding the accountant's point of view in order to develop a joint development of technique. Both individuals must gain an understanding of the other's field in order to develop the basis for a common language.
• Statistical techniques that are used in other areas cannot be blindly accepted for accounting. There is a belief that time will be saved when existing statistical methods can be applied directly. However, it should be noted that there may be certain accounting problems that require new statistical techniques to be developed in order to solve them.
• Satisfactorily operating accounting techniques should not be supplanted by statistical procedures for the sake of change only. An increase in the use of accounting, auditing or management control techniques is the criterion for suggesting integration of present or new statistical techniques in the accounting field.

Further Study on Statistical & Accounting Integration

According to Trueblood and Cooper (1955), the Pittsburgh group found that many of the published statistical applications did not conform to the above mentioned principles. As a result, this group decided to conduct its own studies to support the principles that were discussed earlier. Some of these studies included:

• Internal accounting procedures and management control problems.
• In a small specialty steel manufacturing corporation, quality control charts were developed as a way to investigate performance variances on a daily basis. This experiment also produced practical procedures involving statistical correlation techniques to evaluate the consistency of cost standards.
• A LIFO price index based on statistical sampling was developed to yield reductions in cost, higher quality results and simplication of administrative problems. It was found that the data could be used for other managerial initiatives such as price index purposes. Some of these correlative initiatives included forecasting, establishing cost of sales determinants, and calculation of interim inventory turnover rates.
• Auditing cases.
• The aging of accounts receivable in department stores. The sampling procedures developed for aging purposes are directed toward both the control and audit processes.
• A statistical application was made to the evaluation of the recorded book value of inventory. The evaluation was completed by statistical analyses of adjustments to records developed from past cycle counts.
• An audit application had been made in relation to physical tests of bulk inventories for internal and external audit purposes. Two significant findings of this experiment were: The external auditor decided that his sample should be increased versus decreased, and; Statistical sampling cannot yield significant protection to the auditor in fraud detection without large samples.

Audit Sampling Technique

The American Institute of Certified Public Accountants' (AICPA) auditing standards board (ASB) had formed a task force to develop and implement an audit sampling policy for using the sampling techniques as described in Statement on Auditing Standards (SAS) number 39, Audit Sampling (Journal of Accountancy, 1983). The policy became effective for any examination of financial statements for periods ending on or after June 25, 1983. However, when a meeting was held in April, 1983, the ASB recommended that the AICPA provide additional guidance to practitioners for implementing the provisions. The task force's response was to create a question and answer (Q&A) list with the five most frequently asked questions by practitioners.

In 1992, the Audit Sampling and Analytical Techniques Committee of the New York State Society of CPAs conducted a survey of New York accounting firms. According to Hitzig (1995), the purpose of the survey was to obtain information regarding the use of audit sampling based on SAS No. 39, Audit Sampling. The main interest was to determine the level of use of the audit sampling technique by local accounting firms. A survey had been conducted in 1984 by the Audit Testing Techniques Subcommittee of the AICPA. However, it did not address the use of sampling in practice.

Application

Statistical Sampling

"To meet their clients' needs in an environment of heightened competition and runaway inflation, many auditors are turning to scientifically supported methods of planning, executing and evaluating audit procedures to obtain evidential matter. Statistical sampling is one such method" (Akresh & Zuber, 1981, p. 50). There are many ways that an accountant can set up a statistical sampling. Hitzig (2004) provided a model that worked on the premise that one could set up a statistical sampling by defining the population, frame and sampling unit.

The Population

The population is the set of all accounts or transactions that the auditor wishes to use in order to arrive at the conclusion. The first step in the process is to define the test objective. Once the test objective has been determined, the auditor should define the population. The steps are in this order so that the auditor can draw a sample based on the specific test objective.

The Frame

Once the testing has been completed, the auditor must attribute the results to the items versus the population since auditors do not select a sample directly from the population. This representation is referred to as the frame. The frame provides the auditor with a foundation for identifying items to be included in a sample.

In most cases, the accounting population is presented in the form of a list (i.e. payroll file, accounts receivable detail). This list (or frame) tends to streamline and simplify the sample selection process. However, the population's sampling frame does not have to be a list. Sometimes, the physical locations provided by floor plans or other population identifiers can be used as frames. Also, there may be an occasion where the auditor has to create an appropriate frame when one is not available. Regardless of whether or not a list is used, the selection of a frame is usually based on convenience and accessibility. Most convenient frames are computer data files. If these files are used, there is an opportunity to integrate them with an application of computer-assisted audit techniques and data retrieval.

There are some circumstances where the auditor has to be on alert to make sure that they do not encounter any problems with their samplings. Hitzig (2004) provided some examples such as:

• Over specified frames If there are units in the frame that do not contain members of the population, they are not applicable to the conclusion that the auditor wishes to draw. These units would be considered irrelevant.
• Underspecified or Incomplete frames As the auditor makes plans to collect a sample, he/she must ensure that every item in the population is also in the frame. If a frame is not complete, there is a probability of some significant members of the population not being included in the sample. If this type of action were to occur, there is a violation of AU 350's requirement for representativeness, which requires that every item in the population under examination must have a chance of being selected. If a frame is incomplete, there is an opportunity for biased estimates of the population value that is under examination. This statement is true especially if the auditor is not careful to distinguish between the size of the population and the size of the frame on which the selection of the sample was performed.

The Sampling Unit

A population is composed of basic units that are clustered into sampling units. The sampling unit is determined by the auditor's choice of frame. The item that the auditor conducts the examination on is referred to as the sampling unit, and the sampling unit is vouched or traced. The examination can be conducted by inspection, observation or confirmation.

Auditors tend to make the statistical sampling procedures flexible. If the selection and evaluation are performed properly, there is a high probability that there will not be any questions regarding validity due to technical issues. For example, if the total recorded amount of the sampling units equals the total recorded amount of the population under examination, the technical information has been confirmed. Therefore, it is valid.

How are sampling units selected? Auditors have different preferences. However, listed below are some common trends in the field.

• Accounts. Accounts are the preferred method of sampling unit, especially if dealing with consumer accounts (i.e. credit cards). Using this approach will allow the auditor to directly confirm the net balance in any designated account. However, there may be problems if an auditor attempts to confirm account balances on commercial accounts. Therefore, commercial accounts tend to be maintained on vouchers payable systems.
• Open invoices. If an organization has a file of open invoices that can be directly accessed, open invoices would be the preferred method. Since the open invoices only consist of debits to the accounts receivable, the auditor will need to apply a separate test procedure for credits to accounts.
• Since many organizations document their purchases in a vouchers payable system, they have found that it is easier to confirm individual invoices versus account balances. This choice of sampling unit may be applied with either equal probability (i.e. simple random sampling) or with probability proportional to size (i.e. dollar-unit sampling).
• Invoice line items. Dollar-unit sampling enables an auditor to choose an invoice line item as the sampling unit. This approach is referred to as subsampling (Leslie, Teitlebaum, & Anderson, 1980). In this scenario, a computer program identifies the invoice and the dollar within the invoice in which the selected line item is located. The auditor is responsible for manually identifying the line item by footing the invoice until the selected item is found. The auditor only has to vouch that item, and every other selected line item in the sample. In dollar-unit sampling, the auditor projects the results associated with the selected line items by using the total book value of the frame as the representation of the frame size.

Viewpoint

Audit Sampling

As mentioned earlier, SAS No. 39 was adopted in 1981. Since that time, accounting firms have adapted their policies and procedures to the requirements listed in SAS No. 39. Although SAP No. 54 only included statistical sampling, SAS No. 39 includes statistical and nonstatistical audit sampling. The four key requirements that are related to audit sampling include:

• A sample should be selected in such a way so it may be representative of the population from which it is selected.
• Errors disclosed in a sample should be projected to the population, thus yielding an estimate of the total amount of error in the population.
• The auditor should consider sampling risk, which is the risk a sample will result in an incorrect audit decision.
• The auditor should consider tolerable error, which is the auditor's specification of the largest error that may exist in the sampled population without causing the financial statements to be materially misstated.

Audit sampling did not become popular until the 1970s when many of the large accounting firms decided to invest in the development and delivery of statistical sampling policies and support. Kenneth Stringer, chairman of the Statistical Sampling Subcommittee of the AICPA, and Herbert Arkin, a professor of statistics at Baruch College and a consultant to two Big Eight firms and the IRS, were two of the pioneers in this effort. Stringer used SAP No. 54 to introduce the concept of audit sampling to the accounting field while Arkin wrote the first book on statistical techniques for auditing and it was geared toward practitioners in the profession.

Decline in Statistical Sampling

Unfortunately, the use of statistical sampling as a methodology for audit testing experienced a drastic decline in the 1980s. Many have speculated as to why this decline occurred. Some have implied that the decline was a result of the changed nature of internal control work, which affected the way many firms organize their audit procedures. Some firms, especially those among the Big Six, had reduced or eliminated tests of transactions as a test of controls. The main reason for this change was attributed to the viewpoint that a transaction test provides little or no information as to the performance of control procedures over a routine data process.

Another reason for the decline in statistical sampling was attributed to SAS No. 39. The AICPA's audit guide defines nonstatistical sampling as "any sampling procedure that does not measure the sampling risk". Therefore, random selection and nonstatistical evaluation were considered acceptable under GAAS, which gave equal status under GAAS to both nonstatistical and statistical sampling. As a result, many auditors elected the nonstatistical approaches since they were easier to apply.

At the Financial Executives International (FEI) peer-to-peer forum in 2013, however, participants were in agreement that detailed reporting at the unit level was significantly less useful than higher-level statistical aggregation reports, which facilitated analysis and enabled management decisions. An understanding of profitability was more easily gained when business divisions and lines within those divisions could be grouped and identified by markets, products, operations, and so on, especially when a contribution margin reporting approach was used.

Conclusion

Many have argued that statistics is important to the field of accounting. The tools can assist accountants with being more effective on their job. In addition, "there is definite evidence in accounting periodicals of an increasing interest in the use of statistics, especially statistical sampling techniques, in accounting" (McGurr, 1960, p. 60). Many scholars have recognized that the two fields can yield creative results when they are combined.

There has been much debate among practitioners as to whether or not statistical sampling is useful. There are two main concerns that are mentioned when conversations about the topic are discussed. The first concern deals with the perception of practitioners. Some believe that statistical sampling is not useful in their practice because "it replaces the auditor's judgment with mechanical procedures and because it is difficult to understand and apply" (Akresh & Zuber, 1981, p. 50). The second concern deals with educating practitioners on statistical sampling. According to Akresh and Zuber (1981), some auditors have found it useful to seek the guidance of a statistical sampling specialist. These researchers indicated that some ways of obtaining a specialist include:

• Obtaining the help of a professor from a local university on a consulting basis.
• Hiring an auditor experienced in statistical sampling.
• Developing an internal specialist.
• Participating in a professional group (p. 53).

The American Institute of Certified Public Accountants' (AICPA) auditing standards board (ASB) had formed a task force to develop and implement an audit sampling policy for using the sampling techniques as described in Statement on Auditing Standards (SAS) No. 39, Audit Sampling (Journal of Accountancy, 1983). The policy became effective for any examination of financial statements for periods ending on or after June 25, 1983. However, when a meeting was held in April, 1983, the ASB recommended that the AICPA provide additional guidance to practitioners for implementing the provisions. The task force's response was to create a question and answer (Q&A) list with the five most frequently asked questions by practitioners.

SAS 39 mandates that sample items be selected based on their ability to be representative of the population. In order to ensure that this directive is applied, auditors must understand what populations, frames and samplings are. If the auditor has an understanding of these terms and how they relate to a test of details of an account or a class of transactions, the auditor can properly execute an audit test of details and draw valid, defensible conclusions. Although audit sampling is slowly returning as the basis for the most rigorous test procedure available to an auditor, there is a need to reeducate auditors in the basics of sampling (Hitzig, 2004).

Terms & Concepts

American Institute of Certified Public Accountants: A professional association for CPAs providing guidance to members on accounting techniques and standards.

Audit Sampling: The application of audit procedures to less than 100 percent of the items within a population to obtain audit evidence about a particular characteristic of the population.

Dollar Unit Sampling: A method that uses a combined-attributes-and-variables method of statistical inference. It can be used simultaneously for both variables and attributes sampling. It differs from most sampling techniques in that the sampling units are defined as individual dollars rather than as physical units (such as inventory items). The procedures are performed on the individual accounts or inventory items containing the dollars selected.

Simple Random Sampling: A sample in which the population is first divided into strata (classes of elements). Within each stratum, each element has an equal chance of being chosen for the sample.

Statistical Correlation Technique: In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables.

Statistical Sampling Technique: A method of selecting a portion of a population, by means of mathematical calculations and probabilities, for the purpose of making scientifically and mathematically sound inferences regarding the characteristics of the entire population.

Statement on Auditing Standard: Guidelines used to establish standards and provide guidance on the design and selection of an audit sample and the evaluation of the sample results.

Bibliography

Akresh, A., & Zuber, G. (1981). Exploring statistical sampling. Journal of Accountancy, 151, 50-56. Retrieved August 25, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=4585670&site=ehost-live

Audit sampling task force to aid applying SAS no. 39. (1983). Journal of Accountancy, 156, 12-14. Retrieved August 26, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=17784793&site=ehost-live

Fan, Q., & Zhang, X. (2012). Accounting conservatism, aggregation, and information quality. Contemporary Accounting Research, 29, 38-56. Retrieved October 31, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=74078432&site=ehost-live

Hitzig, N. (1995). Audit sampling: A survey of current practice. CPA Journal, 65, 54-57. Retrieved August 26, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=9507266586&site=ehost-live

Hitzig, N. (2004). Elements of sampling: The population, the frame, and the sampling unit. CPA Journal, 74, 30-33. Retrieved August 26, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=15023175&site=ehost-live

Leslie, D., Teitlebaum, A., & Anderson, R. (1980). Dollar unit sampling: A practical guide for auditors. London: Pitman.

McGurr, F. (1960). The integration of statistics and accounting. The Accounting Review, 35, 60-63. Retrieved August 25, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=7061138&site=ehost-live

Najarian, G. (2013). What's the 'best' approach to profitability management reporting?. Financial Executive, 29, 67. Retrieved October 31, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=85920002&site=ehost-live

Trueblood, R. (1953, October). The use of statistics in accounting control. The New York Certified Public Accountant, 619-626.

Trueblood, R., & Cooper, W. (1955). Research and practice in statistical applications to accounting, auditing, and management control. The Accounting Review, 30, 221-229. Retrieved August 25, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=7060694&site=ehost-live

Suggested Reading

Finley, D. (1989). Decision theory analysis of audit discovery sampling. Contemporary Accounting Research, 5, 692-719. Retrieved August 26, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=10967302&site=ehost-live

Hall, T., Hunton, J., & Pierce, B. (2002). Sampling practices of auditors in public accounting, industry, and government. Accounting Horizons, 16, 125-136. Retrieved August 25, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=6955022&site=ehost-live

Power, M. (1992). From common sense to expertise: Reflections on the prehistory of audit sampling. Accounting, Organizations & Society, 17, 37-62.