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Sampling and small populations Ljubljana, 12-13 October 2009. Henk W. H. Geilen RE RA CISA Senior auditmanager Dutch Audit Authority. Outline. Introduction Why Sampling What is Sampling “Small populations” Conclusion. Why sampling ?. A true Story A long time ago………. - PowerPoint PPT Presentation
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1
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Sampling and small populations Ljubljana, 12-13 October 2009
Sampling and small populations Ljubljana, 12-13 October 2009
Henk W. H. Geilen RE RA CISA Senior auditmanager
Dutch Audit Authority
Henk W. H. Geilen RE RA CISA Senior auditmanager
Dutch Audit Authority
2
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
OutlineOutline
1. Introduction
2. Why Sampling
3. What is Sampling
4. “Small populations”
5. Conclusion
3
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Why sampling ?• A true Story
• A long time ago……….
4
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
5
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
What’s the problem ?• Situation A (no errors)
• What to do?
• Stop?
• Seen Enough?
6
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
What is Sampling (2)•Situation B (errors found)
•What to do?
•Take more items?
•Correcting only the errors found?
•……
7
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
So…..•You never know whether you have seen enough items
•So did you do “enough” work?
• “Enough” means:
•Not to few
•Not to much
8
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Mathematics ?•Total number of balls
•Number of red balls
•Chance : Red /total
•Number of draws
• If you put the balls back after drawing
•Multiply chances for each draw
9
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Mathematics/Formula•Red/(Red + White) * Red/( Red + White)…
•Number of draws: n
•Red / (Red +White) = Red / Total = p
•Chance: ß
•ß = pn
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
So What ?•Audit
•We know “total”
•We don’t know “error” p
•We don’t want to know “Chance” ß
•…..
11
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Audit•The “object” is good
•We mean the “object" is good ENOUGH
•So there can be a little error
•Let’s call it MATERIALITY
•Hé isn’t that p
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Chance•100% assurance
• Is this possible ?
• Is this necessary ?
•The auditor defines his risk
•This means: the chance that he accepts the population while the error is higher than acceptable
13
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Where are we ?• We have a formula• ß = pn
• Let’s say ß = 0,05• So the risk is 5% or the Probability = 95 %• Say the materiality = 1 % (0,01)• Then we can calculate n = the number of samples• 0,05 = (1 - 0,01)n
• N = 300
14
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
This means• If you want to accept 1 percent errors (max)• You sample 300 times• You find zero errors• The risk is (less than) 5 %• or• If you take a sample of 300 items and find 0 errors you know 95 % sure that the maximum error in the population is 1 percent.
15
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Playing with n•Different materiality:
• f.e. 2 percent n = 300/ 2 = 150
• f.e. 0,5 percent n = 300 / 0,5 =600
•Different risk:
•10 percent : n * p = 231
•15 percent : n * p = 190
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Based on 0 errors•Risk 5 percent
•1 error : n * p = 475
•2 errors : n * p = 630
•3 errors : n * p = 776
•4 errors : n * p = 916
•5 errors : n * p = 1052
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Risk•ß (Beta) Risk :
The risk that you accept the population on the basis that the error is less or equal than the materiality
•α (Alpha) Risk :
The risk that you don’t accept the population while in fact there is no error
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
What is important•Random / A-select
•Meaning : each element has an equal change of being sampled
•The sampled set has the same characteristics as the population
19
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
First ConclusionsSampling :
• A method to calculate how much work to do
Parameters :
• Materiality
• Confidence level
• Not size of the population
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Back to small populations• The questions are/stay:
• How many items (operations) to audit
• What items (operations) to audit
• How to audit an item (operation)
• Non statistical sample
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
How can we do it (how much)?• Step 1: Audit all “big” items (big means >
materiality)
• Step 2: Calculate sample size for other items based on results system audit
• This defines the confidence level
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
In detailTotal population (all operations)
Minus Big operationsIs Population to SampleMultiply Confidence LevelIs “temp result”Devide (Population to sample) multiply Materiality
Is Sample Size
23
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Assurance level from systems audit
Reliability Confidence Level
Factor
Only minor improvements
High 60 % 0,92
Some improvements
Average 70 % 1,21
Substantial improvements
Average 80 % 1,61
Does not work Low 90 % 2,31
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Example141.596.219 (393 operations)
Minus 15.312.795 ( 4 operations)Is 126.283.424Multiply 1,21 (70 %)Is 152.802.943Divide 2.525.668 (0.02 *126.283.424 )
Is 60 Sample Size (plus 4)
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Which items (1)• Random• Verify• Has sample same characteristics as
population• Average..• MA, Region etc• Looks like risk model Commission
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
What items (2)• Not the same characteristics =
• Draw again
• After a few draws ….
• Expand sample
27
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
After the auditTotal error fractions
Divide sample sizeIs average error fractionMultiply population to sampleIs projected errorPlus error in big operationsIs Total error
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Evaluation• Calculate materiality
• Compare with total error
• Example
• 0,02 * 141.596.219 = 2.831.924
• Vs 5.096.546
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
What can you influence ?
141.596.219 (393 operations)Minus 15.312.795 ( 4 operations)
Is 126.283.424Multiply 1,21 (70 %)Is 152.802.943Divide 2.525.668 (0.02 *126.283.424 )
Is 60 Sample Size (plus 4)
30
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Reminder• What was the goal?
• Sample?
• Audit?
• Audit Opinion?
31
Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Conclusions• No miracle solutions (Mathematic not Magic)• basic principles equal to statistic sampling• definition of the audit objective• definition of the population• definition of the characteristics to test• define confidence level, materiality• random selection
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
More conclusions• non-statistical sample size rather large to support
audit judgment• consider increasing tests of controls to determine
confidence level for sampling• difficult to determine precision of error projection• representativeness of the sample needs to be
evaluated
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Ljubljana, 12-13 October 2009
Homologues Group Meeting
Slovenia, October 2009
Republika SlovenijaEuropean Union
Thank you for your attention!Thank you for your attention!
Henk GeilenHenk Geilen
[email protected]@rad.nl+31 6 3100 7794+31 6 3100 7794