3 DEMYSTIFYING STATISTICAL SAMPLING The Principle (or Law) of Parsimony: That things are usually connected in the simplest or most economical way. Reducing ideas to small, easy-to-write symbols & saying a lot in a small area covered by a formula. Eliminate the Greek, Arabic & Roman language barrier in symbols & Formulae that mystify Mathematics or Statistics. Just like any other audit, Probe Statistical Assertions-Life can be made easy with appropriate sampling. If it cannot be measured, then it cannot be managed economically, efficiently, & effectively. Mathematics or statistics is commitment to logical thinking. It squeezes the most learning about the population from limited sample data.
4 WHY DO AUDITORS SAMPLE? International Standards for the Professional Practice of Internal Auditing: Guides Information should be: Sufficient, Reliable, Relevant & Useful Acknowledges Sampling Techniques in Evidence Acquisition Opinions are NOT ABSOLUTE GUARANTEE but REASONABLE ASSURANCE of Accuracy Proficiency & Due Professional Care Cost-Benefit Considerations: The Economy, Efficiency & Effectiveness, Corroborating Evidence for Control Processes & Account Balances
5 SAMPLING POLICY Written Policy Statement When to Sample? Who Should Sample? How to Sample? Inappropriate Uses for Sampling: When a Total is easily Audited Inquiry & Observation Procedures Analytical Procedures
6 STATISTICAL & NON-STATISTICAL SAMPLING Three Characteristics in Common: Both Require Auditor judgment in Planning, Implementing, & Evaluating the Sampling Plan Actual Audit Procedures Performed are the same Both Non-Statistical & Statistical Techniques are Permitted by the IPPF
7 STATISTICAL & NON-STATISTICAL SAMPLING Differences between Statistical & Non-statistical Sampling Sampling Risk is Controlled & Measurable Technical Training & Knowledge is Required Computer Accessibility
8 STATISTICAL & NON-STATISTICAL SAMPLING In Summary, the following should be addressed: What is the Internal Audit Department s Recommended Policy or Procedure? Is a Quantitative measure of Sampling Risk Desired? What is the relative Cost & Benefit of Statistical versus Nonstatistical Sampling? Is Technical Expertise Available? Is Computer Software Accessible or Expertise to Write a Program?
9 STATISTICAL TERMINOLOGIES Confidence Level (C): Is the Reliability Level or Degree of Belief in the Obtained Results. Measure of Central Tendency: Mean (µ): The arithmetic average of a set of numbers. Median: The halfway value of raw data arranged in numerical order from lowest to highest. Mode: The most frequently occurring value.
10 STATISTICAL TERMINOLOGIES Standard Deviation (σ): The statistical measurement of the variability of values in a sample (the square root of the variance). Range: The difference between the largest and smallest values of any group. Population (N): The total number of items from which the sample is drawn-it s the focus of interest comprising sampling units. Sampling Unit: Individual items making up a Population. Sample (n): Collection of sampling units on which audit procedures are performed.
11 STATISTICAL TERMINOLOGIES Logical Unit: Account or transaction selected to be sampled. Expected Population Deviation Rate (ρ): Estimate of the actual deviation rate in the population, usually based on prior experience, inquiries, and observations. Precision (P): An assumed amount of possible unknown or the range of allowable error. Tolerable Misstatement: The auditor s assessment of materiality with respect to the population. Upper Precision Limit: Upper limit on deviations expected in the population.
12 STATISTICAL TERMINOLOGIES Tainting: Percentage of misstatement in a logical unit in a PPS sample. Upper Misstatement Limit (UML): Estimated maximum misstatement existing in the population at a specified reliability in PPS sampling. Sampling Risk: Conclusions based on sample differing with conclusions that could be reached if the entire population were examined. Non-sampling risk: Drawing incorrect conclusion for reasons other than sampling due to poor judgment or failure to adhere to professional standards.
13 STATISTICAL SAMPLING Advantages Disadvantages May yield desired results from minimum number of items Yields quantified data Includes measures of sampling risk, confidence level, and precision Is adaptable to computer testing Lends credibility to audit conclusions/recommendations Can be costly and timeconsuming May require training and software costs May preclude experienced auditors insights
14 NON-STATISTICAL SAMPLING Advantages Disadvantages Flexibility Use of internal auditor s judgment Allows reasonable reliability at reasonable cost Results not statistically valid No objective measure of sampling risk provided Chance of wrong sample size Effectiveness depends upon auditor s skill
15 STATISTICAL SAMPLING PLANS 1. ATTRIBUTES SAMPLING (TESTS OF CONTROLS) Concerns binary, yes/no, or error/non-error populations It tests the effectiveness of controls. 2. VARIABLES SAMPLING (SUBSTANTIVE TESTS) Concerns monetary amounts & other measures. It assesses materially misstated account balances & THE PPS SAMPLING ( THE CAV SAMPLING) Concerns primary engagement objective of few overstatements & not understatement. Difference & Ratio Estimations may not be efficient.
16 EXTERNAL AUDITING STANDARDS Internal & External Audit Work Coordination & Recognition: Statement on Auditing Standards (SA) No. 39: Audit Sampling & SAS No. 47: Audit Risk & Materiality in Conducting an Audit AICPA. Audit Risk Model: Audit Risk = Inherent Risk x Control Risk x Detection Risk Audit Risk: Issuing unmodified opinion on financial statements that are materially misstated. Inherent Risk: Material misstatement occurring in the absence of appropriate controls. Control Risk: Controls ineffective & fails to prevent or detect material misstatement in a timely manner. Detection Risk: Substantive procedures failing to detect a material misstatement.
17 EXTERNAL AUDITING STANDARDS Sampling risk impacts the Efficiency & Effectiveness of an audit Components of Sampling Risk Audit Test Audit Efficiency Audit Effectiveness Tests of Controls Risk of Assessing Control Risk Too High (i.e., not depending upon effective controls) Risk of Assessing Control Risk Too Low (i.e., depending upon ineffective controls) Substantive Tests Risk of Incorrect Rejection (i.e., rejecting a materially correct balance) Risk of Incorrect Acceptance (i.e., accepting a materially incorrect balance) Statistical Term Alpha Risk ( ) Beta Risk ( β)
18 EXTERNAL AUDITING STANDARDS Non-sampling Risk The audit failing to detect an internal control weakness or material misstatement for reasons other than the fact that sampling was used. Application of an inappropriate audit procedure Failure to recognize an error condition Omission of an essential audit step Materiality: Amount of difference tolerated by the auditor & concluding the assertion tested as reasonable: Tolerable deviation rate for tests of control Tolerable misstatement for substantive testing Materiality is inversely related to sample size Materiality assessment must be a cost versus benefit decision
19 SAMPLE SELECTION METHODS Methods Appropriate for Both Statistical & Non-statistical Sampling: Simple Random Sampling: Items with equal chance of selection. Systematic Sampling: nth item selection with random start within the n interval. PPS uses systematic sampling. Methods Used Only for Non-statistical Sampling: Haphazard Selection: Selecting sample items without intentional bias. Block Selection: Audit of a group of contiguous transactions like delivery notes for March or invoices in a sequence. Block Amount: Whole amount is audited. Other Considerations in Sample Selection: Void Items: Select additional sampling units for voided items. Missing Items: Must be treated as an error condition- In attributes, control is not effective & in substantive testing, audited value is ZMK 0.00
20 ATTRIBUTE SAMPLING When to use. Size of sample (n) Statistical table specifications To estimate the number of times a certain characteristic may occur in a population Based on judgment about probability that errors (or other characteristics) will occur or based on statistical tables n = C2 ρq P 2 Population size (N) Confidence level (C) Precision (P) Expected rate of errors (ρ) &q=100-ρ
21 Attributes Sampling Illustrations ITEM ACCOUNTS RECEIVABLES AS AT 31ST DECEMBER Population Size of Accounts Receivable N 4,000 Accounts 2 Confidence Level 90% Confidence Coefficient C Tolerable Deviation Rate (TDR) 5% (Based on Prior Years of Findings or Pilot Sample) 4 Planned Risk of Assessing Control Risk Too Low (Beta Risk) β 5% 5 Planned Risk of Assessing Control Risk Too High (Alpha Risk) 10% 6 Desired Precision = Beta x TDR/Alpha P 2.50% 7 Sample Size n 204 Accounts 8 Expected Number of Errors (From Statistical Tables) 5 Assuming Control Procedures Anticipated Deviation Rate = Zero 0% Upper Precision Limit (UPL) from the Statistical Tables 1.50% (And is Less than Tolerable Deviation Rate=5%) 9 Assuming 2 Actual Control Procedure Errors: 2 Upper Precision Limit (from the Tables) UPL 3.20% 10 And UPL < ρ Conclusion??? 11 CONCLUSION Controls are Effective
22 Attributes Sampling Variations Stop-or-Go Sampling: The Auditor guards against selecting an unnecessarily large sample. Discovery Sampling: The Auditor targets discovering at least one deviation if the percentage of deviations in the population is at or above a specified level, e.g. Fraud, Substantial mistake or Compliance failure.
23 VARIABLES SAMPLING When to use. Size of sample (n) Statistical table specifications When size matters; e.g., amount of a discrepancy in monetary or weight terms n = C2 σ 2 P 2 Population size (N) Confidence level/coefficient (C) Precision (P) Standard deviation (σ)
24 Variables Sampling Illustration ITEM ACCOUNTS RECEIVABLES AS AT 31ST DECEMBER Recorded Amount of Accounts Receivable (N) RM 360,000 ZMK 2 Tolerable Misstatement TM 18,000 ZMK 3 Planned Risk of Incorrent Acceptance (Beta Risk) β 5% 4 Planned Risk of Incorrect Rejection (Alpha Risk) 10% 5 Number of Accounts Receivable (N) 4,000 Accounts 6 Estimated Population Standard Deviation 8.68 ZMK (Based on Prior Years of Findings or Pilot Sample) 7 Confidence Level 90% Confidence Coefficient C Desired Precision = Beta x TM/Alpha 9,000 ZMK 9 Precision per-item basis (Desired Precision/N) P 2.25 ZMK 7 Sample Size n 40 Accounts
25 Three Types of Variables Sampling Mean-per-unit Estimation: Estimates the total monetary amount of the population by calculating a sample mean & multiplying by the number of items in the population. Difference Estimation: Estimates the total error in the population. Useful only if population contains enough errors to generate a reliable sample estimate & the differences are not proportional to the book values. Ratio Estimation: Estimates the total monetary amount of the population by calculating the ratio between the audited & book values in the sample and using this ratio to make the estimate. Useful when differences between book & sample values are proportional to book values.
26 Variables Sampling: Mean-per-Unit Estimation Case example Population: 4,000 Accounts Total book value: ZMK 360, Sample size: 40 Accounts Sample book value: ZMK 3, Sample audit value: ZMK 3, Step 1: Calculate average audit value (i.e., mean-perunit value for audited samples). K3,400.00/40 = K85.00 / Account. Step 2: Multiply mean-per-unit value by number of accounts in the population. K ,000 Accounts = K340, Over-count = K20, (K340, K360,000.00)
27 Variables Sampling: Difference Estimation Case example Population: 4,000 Accounts Total book value: ZMK 360, Sample size: 40 Accounts Sample book value: ZMK 3, Sample audit value: ZMK 3, Step 1: Calculate average difference between audit value and book value for the sample. (K3, K3,600.00)/40 Accounts = (K5.00) Step 2: Determine the difference estimate for the population. (K5.00) 4,000 accounts = (K20,000.00) Step 3: Estimate actual value by adding the difference estimate and book value for the population. (K20,000.00) + K360, = K340, Book value is Overstated by K20,000.00
28 Variables Sampling: Ratio Estimation Case example Population: 4,000 Accounts Total book value: ZMK360, Sample size: 40 Accounts Sample book value: ZMK3, Sample audit value: ZMK 3, Step 1: Audit value for sample = K3, Step 2: Book value for sample = K3, Step 3: Find ratio of audit value to book value: K3, / K3, = 0.94 Step 4: Estimate actual population value by multiplying ratio by population book value: 0.94 K360, = K338, Book value is Overstated by K21,600.00
29 PROBABILITY-PROPORTIONAL-TO-SIZE (PPS) SAMPLING When to use Size of sample (n) (n1: AM=0, & n2: AM>=1) Statistical specifications When auditing. account balances for few overstated items; e.g., in inventory, receivables, disbursements, etc. n1 = RM x RF TM or n2 = Recorded Amount of the Account (RM) Reliability Factor (RF) Tolerable Misstatement (TM) Anticipated Misstatement (AM) Expansion Factor (EF) RM x RF TM (AM x EF)
30 PPS ILLUSTRATION ACCOUNTS RECEIVABLE AS AT 31ST DECEMBER Recorded Amt of A/C Receivables RM 360,000 Tolerable Misstatement TM 18,000 Anticipated Misstatement AM 0 Risk of Incorrect Acceptance 5% A/C No. AMT ZMK CUM AMT Kwacha Selected Sampling Unit Observed Amount Tainting % Sampling Interval Projected Misstatement ACT0001 9,450 9,450 9,000 9,450 7,875 * * 1,575 ACT ,930 ACT0003 2,800 12,730 ACT0004 5,106 17,836 ACT0005 2,100 19,936 18,000 2, % 9,000 9,000 ACT0006 8,000 27,050 27,000 8,000 8, , ACT4000 6, , ,000 6,000 4,500 25% 9,000 2,250 TOTAL 360,000 12,825 Basic Precision(SI x RF = K9,000 x 3) ZMK 27,000 Total Projected Misstatment ZMK 12,825 Allowance for Precision Gap Widening: ( ) x K9,000 ZMK 6,750 ( ) x K2,250 ZMK 1,238 Upper Misstatement Limit (UML)>TM ZMK 47,813 CONCLUSION Accounts Receivable Materially Overstated
31 CONCLUSION/RECOMMENDATIONS It is Concluded & Recommended that Internal Auditors comply with the Proficiency & Due Professional Care IIA Standards by Appropriate Application of both Statistical & Non Statistical Sampling to Reasonably Assure that Opinion Evidence is: Sufficient, Reliable, Relevant and Useful.
32 REFERENCES FOR FURTHER READING 1. Sampling for Internal Auditors: Text-based Self Study Course- The Institute of Internal Auditors by Barbara Apostolou, PhD, CPA, DABFA. 2. Internal Audit Practice-Part 1: The IIA s CIA Learning System by The Institute of Internal Auditors. 3. Internal Audit Practice-Part 1: Gleim CIA Review by Professor Irvin N. Gleim, PhD, CPA, CIA, CMA, CFM.
33 COMMENTS, REMARKS & QUESTIONS
34 Confidence coefficient, C, Based on the Risk of Incorrect Rejection Risk of Incorrect Rejection Confidence Level Confidence Coefficient 20% 80% % 90% % 95% % 99% 2.58
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