RELATING STATISTICAL SAMPLING TO AUDIT OBJECTIVES

1 A STATISTICAL SAMPLING plan, designed especiallyfor auditors , that would help reducenonsampling error by organizing available datainto a quantifiable form for decision-makingRELATING STATISTICAL SAMPLINGTO AUDIT OBJECTIVESBY ROBERT K. ELLIOTT AND JOHN R. ROGERSFOR many years independent auditors have beenencouraged to use STATISTICAL SAMPLING pro-cedures in conducting AUDIT tests. The advantageshave been pointed out in many books and articles onthe subject. Most auditors trained in the last decadehave been exposed to STATISTICAL SAMPLING duringtheir formal training and encouraged to use it. TheAmerican Institute of Certified Public Accountantshas encouraged its use. Notwithstanding all of thisencouragement, use of STATISTICAL techniques is stillnot widespread within the profession. Althoughmany factors contribute to the lagging implementa-tion of STATISTICAL techniques, two stand out as mostimportant;1.

2 STATISTICAL techniques have generally been cum-bersome to apply STATISTICAL techniques have not been adequatelyrelated to common AUDIT the general availability of eomputers inauditing, the cumbersome aspects of statisticalsampling procedures need no longer be done manu-ally. The eomputer will handle the intricacies ofsample size calculation, sample selection and evalu-ation effortlessly and without mechanical of STATISTICAL techniques will vir-tually eliminate the first major deterrent to the useof STATISTICAL SAMPLING in , the second major deterrent must stillbe eliminated before STATISTICAL SAMPLING can bewidely implemented. TTie purpose of this article isto suggest an approach to RELATING STATISTICAL tech-niques to AUDIT OBJECTIVESThe overall objective of an auditor in carryingout the attest function is to make reasonably certainthat financial statements examined by him are ma-terially correct.

3 ^ He tlierefore wishes to find mate-rial errors in the financial statements if they do this he employs numerous procedures, one ofthe most important being the examination of docu-mentary evidence. auditors long ago abandonedany attempt to examine documentary evidence foreach and every item represented in the financialstatements. The resulting partial (or sample) exam-ination must introduce an element of uncertaintyinto his conclusions, for he must draw inferencesfrom incomplete information. This is true whetheror not the auditor uses STATISTICAL SAMPLING uncertainty resulting from partial examina-tion may fnistrate the auditor's desire for materiallycorrect financial statements in two distinct ways:1. Financial statements may be correct, but he mayerroneously conclude they are incorrect andeither qualify his opinion or unnecessarily insistupon Financial statements may be materially in error,but he may erroneously conclude that they arecorrect and give an unqualified financial statements may be in error byany amount (from zero on up), it is only necessaryto control AUDIT risks at the points of zero error andmaterial error (that is, exactly the amount specifiedby the auditor as material) in order to meet theoverall AUDIT objective.

4 Therefore, for the remain-der of this article, only these two magnitudes oferror will be considered. The level of risk at all othermagnitudes ean be observed in Figure 1, page 51."Materially correct" in this context means "fairly pre-sented in conformity with generally accepted accountingprinciples."46 THE JOURNAL OF ACCOUNTANCY, JULY 1972 For convenience, the risk of rejecting perfectly cor-rect financial statements will be referred to as the a(alpha) risk, and the risk of accepting financialstatements in error by exactly a material amountwill be referred to as the f3 (beta) two risks exist even when the auditor usesjudgment SAMPLING procedures, but they cannot inany way be quantified. The result is that, in somecases, auditors do less work than necessary to con-trol these risks and, in other cases, more work thannecessary.

5 When too little work is done, the auditoris not efiFeetively carrying out the attest too much work is done, the auditor is carry-ing out the attest function effectively, but at anexcessive OF STATISTICAL TECHNIQUESS tatistical SAMPLING procedures are ideally suitedto measure the a and /? risks inherent in partial (ortest) examinations. Tf the auditor will state what aand p risks are justified, and what amount he con-siders material for a given account, a STATISTICAL testcan be designed to meet these eriteria. The abilityto measure, in this case, means there is an ability the auditor performs a documentary exam-ination, he may have either or both of two objec-tives:1. To establish the effectiveness of systems and pro-cedures, in order to plan the type, extent andtiming of other AUDIT To establish the material correctness of a finan-cial statement relevant test of material correctness must bestated in dollar terms.

6 Statistically, it must be a testROBERT K. ELLIOTT, CPA, is manager inthe executive office of Peat, Marwick, Mit-chell ir Co. in New York City. He is amember of the AICPA and the Pennsyl-vania Institute of CPAs. His article, "As-pects of Behavioral Accounting" was pub-lished in Statements in Quotes in the April197! JOURNAL. Mr. Elliott received hisbachelor's degree from Harvard Collegeand his degree from Rutfiers University. JOHN CPA, is a partner of Peat, Mar-wick, Mitchell ir Co., in New. York . serves on the Institute's committee onstatistical mmpUng and is a member of theNew York State Society of CPAs and theCanadian Institute of Chartered Account-ants. Mr. Rogers received his bachelor'sdegree from Dartmouth College and degree from Amos Tuck School ofBusiness variables.

7 An attribute test, conversely, can ex-press conclusions only in terms of rates of occur-renee, not attribute test can indicate, for example, thatthe error rate in eertain documents is probably nomore than a certain percentage. While this may beof some use to the auditor, it is not a conclusion thatcan be measured in terms of financial statementimpact. A low error rate does not necessarily meanthat the financial statements are materially correct,nor does a high error rate mean the opposite. Theprincipal usefulness of the technique is to assistthe auditor in making qualitative judgments aboutthe operation of internal controls and aceountingsystems. These judgments may then be used inplanning the type, extent and timing of other auditprocedures, but they cannot normally be used fordirect AUDIT conelusions conceming fair presenta-tion of the financial a STATISTICAL attribute test may be ofsome use in examining systems and controls, it is byno means necessary.

8 Generally, in examining sys-tems the auditor seeks a qualitative conclusion (forexample, the controls over cash disbursements aregood). Seldom does he need precise measurementof rates of occurrence, such as would be providedby an attribute test. Neither is he overly concernedw^ith a and ^ risk the auditor needs quantifiable conclusions,though, he will be best served by a variables test(usually one in which the results are stated in dol-lars). Therefore, the remainder of this artiele willconcern itself solely with variables tests designedto establish the material correctness of financialstatement , there does not appear to be anypublished literature which deals explicitly with theuse of STATISTICAL to measure and eontrolthe auditor's a and P risks with respeet to variablestesting.

9 The AICPA programed instruction series(An Auditor's Approach to STATISTICAL SAMPLING ,Volumes I through IV) and all known textbooksand articles known to us on AUDIT of variablessampling refer to and use variables estimation tech-niques. They propose that the auditor sample a pop-ulation at a given confidence and precision to esti-mate the true , the auditor is advised to relate confi-dence to the level of internal control and precisionto materiality, but he is not given guidanee inplaeing numerical values on confidence and preci-sion. Furthermore, he is not given any advice onwhat to do with the STATISTICAL results. These arecorrectly asserted to be auditing, not STATISTICAL ,matters. However, without some assistance in un-THE JOURNAL OF ACCOUNTANCY, JULY 197247derstanding how STATISTICAL inferences can be re-lated to auditing OBJECTIVES , most auditors are re-luctant to use STATISTICAL techniques.

10 Even whenthey are used, an auditor is not able explicitly tocontrol his a and fi risks. That is, an auditor whohas statistically tested $10 million in accounts re-ceivable and has 95 per cent confidence that thetrue value is between $9 million and $9,500,000 hassome useful AUDIT information, but, based on thestandard approaeh, he cannot come to any eonclu-sion as to what a and ^ risks he has the absence of guidance, many auditors revertto rules of thumb in applying STATISTICAL following approaeh is t>-pieal: let precision(P) equal the amount considered material (M) foran account; let the confidence level (CL) be 95 percent if internal eontrol is poor, or less (say 80 percent) if internal control is excellent; if the statisticaleonfidence interval- includes the book value, acceptit as correct, otherwise reject the book the auditor consistently follows such a plan,with its predefined decision rule, it is possible tocompute the a and ^ risks he is assuming.