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Module G
Variables Sampling
“USA Today has come out with a new survey-apparently three out of every four people make up 75% of the population.”
– David Letterman
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Learning Objectives
Define variables sampling and understand when variables sampling is used in the audit examination.
Understand the basic process underlying probability proportional to size (PPS) sampling as well as when PPS sampling should be used.
Identify the factors affecting the size of a PPS sample and calculate the sample size for a PPS sampling application.
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3. Learning Objectives (Continued)
Evaluate the sample results for a PPS sample by calculating the projected misstatement, incremental allowance for sampling risk, and basic allowance for sampling risk.
Understand the basic process underlying classical variables sampling as well as the use of classical variables sampling in an audit examination.
Understand the basic process underlying nonstatistical sampling as well as the use of nonstatistical sampling in an audit examination.
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Variables Sampling
Used to estimate the amount (or value) of a population
Substantive procedures
Estimate the account balance or misstatement
Compare account balance or misstatement to recorded amount or tolerable error
Types of variables sampling approaches
Probability proportional to size (PPS) sampling
Classical variables sampling
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5. Probability Proportional to Size (PPS) Sampling
Defines the sampling unit as individual dollar in an account balance
Auditor will select individual dollars for examination
Auditor will verify entire “logical unit” containing the selected dollar
Accounts receivable: Customer account
Inventory: Inventory item
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6. Advantages of PPS Sampling
Results in smaller sample sizes
Includes transactions or components reflecting larger dollar amounts
Effective for overstatement errors
Generally simpler to use than classical variables sampling
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7. Disadvantages of PPS Sampling
Provides a conservative (higher) estimate of misstatement
Not effective for understatement or omission errors
Expanding a PPS sample is difficult if the initial conclusion is to reject account balance
Requires special consideration for accounts with zero or negative balances
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8. Major Steps in Variables Sampling
Determine the objective
Define characteristic of interest
Define the population
Determine sample size
Select the sample
Measure sample items
Evaluate sample results
Planning
Performing
Evaluating
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9. Major Steps in Variables Sampling: Planning
Determine the objective of sampling
Determine whether account balance is fairly stated under GAAP
Define characteristic of interest
Instance in which audited value of component differs from recorded value
Define the population
PPS sampling: Dollars comprising account balance
Classical variables sampling: Individual components or transactions comprising account balance
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10. Major Steps in Variables Sampling: Performing
Determine sample size
Sampling risk (risk of incorrect acceptance)
Expected error
Tolerable error
Population size (recorded account balance)
Select sample items
Measure sample items
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11. Effect of Factors on Sample Size in PPS Sampling
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12. Sampling Risks in Variables Sampling
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PPS Sampling Formulae
Sample size
Recorded Balance x Reliability Factor
Tolerable Error - (Expected Error x Expansion Factor)
Sampling Interval
Population Size (Recorded Balance)
Sample Size
Controls exposure to
Sampling risk
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14. Reliability and Expansion Factors
Risk of Incorrect Acceptance
1% 5% 10% 15% 20%
Reliability Factors
(0 Errors)
Expansion Factors
When incorporated into formula, factors are consistent with inverse relationship between risk of incorrect acceptance and sample size.
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15. Major Steps in Variables Sampling: Performing
Determine sample size
Select sample items
Select random starting point in population
Bypass number of dollars equal to sampling interval
Select dollar and identify entire logical unit containing selected dollar
Large items may account for more than one selection
Measure sample items
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16. Major Steps in Variables Sampling: Performing
Determine sample size
Select sample items
Measure sample items
Perform appropriate substantive procedure
Calculate actual misstatement
Audited value – Recorded value
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17. Major Steps in Variables Sampling: Evaluating
Evaluate sample results
Problem with actual misstatement is that it may result from a nonrepresentative sample
Need to “adjust” actual misstatement to control for the risk of incorrect acceptance
Calculate an Upper Error Limit
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Upper Error Limit
Has a (1 – risk of incorrect acceptance) probability of equaling or exceeding the true amount of misstatement
A (risk of incorrect acceptance) probability exists that the true amount of misstatement exceeds the upper error limit
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Upper Error Limit
If upper error limit is $50,000 and risk of incorrect acceptance is 5%
There is a 5% probability that the true misstatement exceeds $50,000
There is a 95% probability that the true misstatement is less than or equal to $50,000
95% 5%
$0 $50,000
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20. Components of Upper Error Limit
Projected misstatement
Incremental allowance for sampling risk
Basic allowance for sampling risk
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21. Projected Misstatement
Assumes the entire sampling interval contains the same percentage of misstatement as the item examined by the auditor
Tainting % = Amount of Misstatement
Recorded Balance of Item
Projected = Sampling Interval x Tainting %
Misstatement
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22. Incremental Allowance for Sampling Risk
“Adjusts” the projected misstatement to control auditor’s exposure to sampling risk
Procedure
Rank all projected misstatements less than sampling interval in descending order
Determine incremental reliability factor for each misstatement
Multiply projected misstatement by incremental reliability factor minus 1.00
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23. Basic Allowance for Sampling Risk
Provides a statistical measure of the misstatement that may be included in sampling intervals in which a misstatement was not detected
Basic Allowance for Sampling Risk = Sampling interval x Reliability factor
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24. The Auditor’s Decision
Upper error
limit 
Tolerable
error
Accept account balance as fairly recorded
Upper error limit
>
Conclude that account balance is not fairly recorded
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25. Classical Variables Sampling (Sample Size)
N x [R(IR) + R(IA)] x SD 2
TE - EE
Differences from PPS sampling
Standard deviation
Risk of incorrect rejection
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26. Reliability Factors for Classical Variables Sampling
Sampling Risk
1% 5% 10% 15% 20%
Risk of Incorrect
Acceptance
Rejection
When incorporated into formula, factors are consistent with inverse relationship between sampling risk and sample size.
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27. Classical Variables Sampling (Evaluating Results)
Precision: Closeness of a sample estimate to the true value
Precision Interval
Estimate Estimate Estimate
- Precision Precision
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28. Classical Variables Sampling (Evaluating Results)
Precision interval has a (1 – risk of incorrect acceptance) probability of including the true balance
Decision
If recorded account balance falls within the precision interval, accept as fairly recorded
If recorded account balance falls outside of the precision interval, conclude that the account is misstated
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29. PPS Sampling vs. Classical Variables Sampling
PPS is more appropriate when:
Overstatement errors are of greater concern
The standard deviation is difficult or impractical to estimate
No or few misstatements are anticipated
The auditor wishes to begin sampling during an interim period
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30. PPS Sampling vs. Classical Variables Sampling
Classical variables sampling is more appropriate when:
The auditor is concerned with both overstatement and understatement errors
The standard deviation can be estimated
Some levels of misstatement are anticipated
The auditor does not begin sampling until after year-end
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31. Nonstatistical Sampling
Does not control the auditor’s exposure to sampling risk
Permitted under generally accepted auditing standards
Differences
Does not consider sampling risk in determining sample size or evaluating sample results
May use a nonprobabilistic selection technique (block or haphazard selection)
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32. Basic Procedure for Nonstatistical Sampling
Select sample
Does not explicitly consider sampling risk in determining sample size
May use block or haphazard selection methods
Measure sample items
Evaluate sample results
Does not consider sampling risk in projected results to population
Compare determined misstatement to tolerable error
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