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By
Mwenya P. Chitalu CIA
SAMPLING FOR EFFECTIVE INTERNAL AUDITING
EXPECTED PRESENTATION OUTCOMES
Why Do Auditors Sample?
Sampling Policy
Statistical & Non-statistical Sampling
Statistical Terminologies
Statistical Sampling Plans
External Auditing Standards
Sample Selection Methods
Illustrations
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.
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
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
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
STATISTICAL & NON-STATISTICAL SAMPLING
Differences between Statistical & Non-statistical Sampling
Sampling Risk is Controlled & Measurable
Technical Training & Knowledge is Required
Computer Accessibility
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 Non-
statistical Sampling?
Is Technical Expertise Available?
Is Computer Software Accessible or Expertise to Write a Program?
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.
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.
STATISTICAL TERMINOLOGIES
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.
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.
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 time-
consuming
May require training and
software costs
May preclude experienced
auditors’ insights
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
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 & ...
3. THE PPS SAMPLING ( THE CAV SAMPLING)
Concerns primary engagement objective of few overstatements & not
understatement.
Difference & Ratio Estimations may not be efficient.
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: 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.
Audit Risk = Inherent Risk x Control Risk x Detection Risk
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)
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
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
ATTRIBUTE SAMPLING
.
When to use
Size of sample (n)
Statistical table
specifications
Based on judgment about probability that errors (or other characteristics) will occur or based on statistical tables
𝐧 =𝐂𝟐𝝆𝒒
𝐏𝟐
• Population size (N)
• Confidence level (C)
• Precision (P)
• Expected rate of errors (𝝆) &q=100-𝝆
To estimate the number of times a certain characteristic may occur in a population
Attributes Sampling Illustrations
. ITEM ACCOUNTS RECEIVABLES AS AT 31ST DECEMBER 2013
1 Population Size of Accounts Receivable N 4,000 Accounts
2 Confidence Level 90%
Confidence Coefficient C 1.64
3 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) 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
𝛽
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.
VARIABLES SAMPLING
.
When to use
Size of sample (n)
Statistical table
specifications
𝐧 =𝐂𝟐𝝈𝟐
𝐏𝟐
• Population size (N)
• Confidence level/Coefficient (C)
• Precision (P)
• Standard deviation (𝝈)
When size matters; e.g., amount of a
discrepancy in monetary or weight terms
Variables Sampling Illustration
. ITEM ACCOUNTS RECEIVABLES AS AT 31ST DECEMBER 2013
1 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) 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 1.64
8 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
𝛽
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.
Variables Sampling:
Mean-per-Unit Estimation
Step 2: Multiply mean-per-unit value by number of
accounts in the population.
Step 1: Calculate average audit value (i.e., mean-per-
unit value for audited samples).
K85.00 4,000 Accounts = K340,000.00
K3,400.00/40 = K85.00 / Account.
Over-count = K20,000.00
(K340,000.00 – K360,000.00)
Case example Population: 4,000 Accounts
Total book value: ZMK 360,000.00
Sample size: 40 Accounts
Sample book value: ZMK 3,600.00
Sample audit value: ZMK 3,400.00
Variables Sampling: Difference Estimation
(K20,000.00) + K360,000.00 = K340,000.00
Step 3: Estimate actual value by adding the difference
estimate and book value for the population.
Step 1: Calculate average difference between audit value and
book value for the sample.
(K3,400.00 – K3,600.00)/40 Accounts = (K5.00)
Step 2: Determine the difference estimate for the
population.
(K5.00) 4,000 accounts = (K20,000.00)
Book value is Overstated by K20,000.00
Case example Population: 4,000 Accounts
Total book value: ZMK 360,000.00
Sample size: 40 Accounts
Sample book value: ZMK 3,600.00
Sample audit value: ZMK 3,400.00
Variables Sampling:
Ratio Estimation
Step 4: Estimate actual population value by multiplying
ratio by population book value:
K3,400.00 / K3,600.00 = 0.94
Step 1: Audit value for sample = K3,400.00
Step 2: Book value for sample = K3,600.00
Step 3: Find ratio of audit value to book value:
0.94 K360,000.00 = K338,400.00
Book value is Overstated by K21,600.00
Case example Population: 4,000 Accounts
Total book value: ZMK360,000.00
Sample size: 40 Accounts
Sample book value: ZMK3,600.00
Sample audit value: ZMK 3,400.00
PROBABILITY-PROPORTIONAL-TO-SIZE (PPS) SAMPLING
. When to use
Size of sample (n)
(n1: AM=0, & n2:
AM>=1)
Statistical
specifications
𝐧𝟏 =𝐑𝐌 𝐱 𝐑𝐅
𝐓𝐌 or 𝐧𝟐 =
𝐑𝐌 𝐱 𝐑𝐅
𝐓𝐌−(𝐀𝐌 𝐱 𝐄𝐅)
• Recorded Amount of the Account (RM)
• Reliability Factor (RF)
• Tolerable Misstatement (TM)
• Anticipated Misstatement (AM)
• Expansion Factor (EF)
When auditing account balances for few
overstated items; e.g., in inventory,
receivables, disbursements, etc.
PPS ILLUSTRATION
. ACCOUNTS RECEIVABLE AS AT 31ST DECEMBER 2013
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
ACT0002 480 9,930
ACT0003 2,800 12,730
ACT0004 5,106 17,836
ACT0005 2,100 19,936 18,000 2,100 0 100% 9,000 9,000
ACT0006 8,000 27,050 27,000 8,000 8,000 0 9,000 0. . . . .. . . . .. . . . .
ACT4000 6,000 360,000 360,000 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:
(4.75-3.00-1.00) x K9,000 ZMK 6,750
(6.30-4.75-1.00) x K2,250 ZMK 1,238
Upper Misstatement Limit (UML)>TM ZMK 47,813
CONCLUSION Accounts Receivable Materially Overstated
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.
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.
COMMENTS, REMARKS & QUESTIONS
Confidence coefficient, C,
Based on the Risk of Incorrect
Rejection
Risk of
Incorrect
Rejection
Confidence
Level
Confidence
Coefficient
20% 80% 1.28
10% 90% 1.64
5% 95% 1.96
1% 99% 2.58
Attributes Sample Size Statistical Tables
For Tests of Controls
Five Percent (5%) Risk of Assessing Control Risk Too Low
(Number of Expected Errors in parentheses)
. Expected
Population
Deviation
Rate (%)
Tolerable Deviation Rate
2% 3% 4% 5% 6% 7% 8% 9% 10% 15% 20%
0.00 149(0) 99(0) 74(0) 59(0) 49(0) 42(0) 36(0) 32(0) 29(0) 19(0) 14(0)
0.25 236(1) 157(1) 117(1) 93(1) 78(1) 66(1) 58(1) 51(1) 46(1) 30(1) 22(1)
0.50 * 157(1) 117(1) 93(1) 78(1) 66(1) 58(1) 51(1) 46(1) 30(1) 22(1)
0.75 * 208(2) 117(1) 93(1) 78(1) 66(1) 58(1) 51(1) 46(1) 30(1) 22(1)
1.00 * * 156(2) 93(1) 78(1) 66(1) 58(1) 51(1) 46(1) 30(1) 22(1)
1.25 * * 156(2) 124(2) 78(1) 66(1) 58(1) 51(1) 46(1) 30(1) 22(1)
1.50 * * 192(3) 124(2) 103(2) 66(1) 58(1) 51(1) 46(1) 30(1) 22(1)
1.75 * * 227(4) 153(3) 103(2) 88(2) 77(2) 51(1) 46(1) 30(1) 22(1)
2.00 * * * 181(4) 127(3) 88(2) 77(2) 68(2) 46(1) 30(1) 22(1)
2.25 * * * 208(5) 127(3) 88(2) 77(2) 68(2) 61(2) 30(1) 22(1)
2.50 * * * * 150(4) 109(3) 77(2) 68(2) 61(2) 30(1) 22(1)
2.75 * * * * 173(5) 109(3) 95(3) 68(2) 61(2) 30(1) 22(1)
3.00 * * * * 195(6) 129(4) 95(3) 84(3) 61(2) 30(1) 22(1)
3.25 * * * * * 148(5) 112(4) 84(3) 61(2) 30(1) 22(1)
3.50 * * * * * 167(6) 112(4) 84(3) 76(3) 40(2) 22(1)
3.75 * * * * * 185(7) 129(5) 100(4) 76(3) 40(2) 22(1)
4.00 * * * * * * 146(6) 100(4) 89(4) 40(2) 22(1)
5.00 * * * * * * * 158(8) 116(6) 40(2) 30(2)
6.00 * * * * * * * * 179(11) 50(3) 30(2)
7.00 * * * * * * * * * 68(5) 37(3)
Attributes Sample Evaluation Tables
For Tests of Controls Upper Limits at Five Percent (5%) Risk of Assessing Control Risk Too Low
. Sample
Size
Actual Number of Deviations Found
0 1 2 3 4 5 6 7 8 9 10
25 11.3 17.6 * * * * * * * * *
30 9.5 14.9 19.6 * * * * * * * *
35 8.3 12.9 17.0 * * * * * * * *
40 7.3 11.4 15.0 18.3 * * * * * * *
45 6.5 10.2 13.4 16.4 19.2 * * * * * *
50 5.9 9.2 12.1 14.8 17.4 19.9 * * * * *
55 5.4 8.4 11.1 13.5 15.9 18.2 * * * * *
60 4.9 7.7 10.2 12.5 14.7 16.8 18.8 * * * *
65 4.6 7.1 9.4 11.5 13.6 15.5 17.4 19.3 * * *
70 4.2 6.6 8.8 10.8 12.6 14.5 16.3 18.0 19.7 * *
75 4.0 6.2 8.2 10.1 11.8 13.6 15.2 16.9 18.5 20.0 *
80 3.7 5.8 7.7 9.5 11.1 12.7 14.3 15.9 17.4 18.9 *
90 3.3 5.2 6.9 8.4 9.9 11.4 12.8 14.2 15.5 16.8 18.2
100 3.0 4.7 6.2 7.6 9.0 10.3 11.5 12.8 14.0 15.2 16.4
125 2.4 3.8 5.0 6.1 7.2 8.3 9.3 10.3 11.3 12.3 13.2
150 2.0 3.2 4.2 5.1 6.0 6.9 7.8 8.6 9.5 10.3 11.1
200 1.5 2.4 3.2 3.9 4.6 5.2 5.9 6.5 7.2 7.8 8.4
Reliability Factors (RF) for Overstatements
.
Number of
Overstatements
Risk of Incorrect Acceptance
1% 5% 10% 15% 20%
0 4.61 3.00 2.31 1.90 1.61
1 6.64 4.75 3.89 3.38 3.00
2 8.41 6.30 5.33 4.72 4.28
PPS Sampling Expansion Factors
For Expected Misstatements
. Risk of Incorrect
Acceptance (%)
Expansion
Factor
1 1.90
5 1.60
10 1.50
15 1.40
20 1.30
25 1.25
30 1.20
37 1.15
50 1.10