Sampling Design Issues in Sales
and Use Tax Audits
This document outlines sampling issues to be considered by auditors
and taxpayers in designing a sales and use tax audit. These issues were
identified from my own review of the literature and helpful comments from
sources in universities, government agencies, industrial firms, consulting
firms, and professional organizations. All views and errors in this document
are my own. I welcome your comments suggestions for improving this document.
Please email any comments or suggestions to wyancey@acrlsbs.com
Written by
Will Yancey, PhD, CPA
38705 Seven Mile Road, Suite 460
Livonia, Michigan 48152
Phone 734.744.4400
Fax 734.744.4150
Email: wyancey@aclrsbs.com
Web: http://www.willyancey.com/
Revised October 7, 2010.
Web address of this document: http://www.willyancey.com/samplingdesign.htm
Contents
1.0 Audit Objectives
2.0 Sampling Methods
3.0 Planning a Statistical Sample
4.0 Executing Sampling Plan
5.0 Evaluating Sample Results
6.0 Projecting Results
Bibliography at http://www.willyancey.com/sampling.htm
1.0 Audit Objectives
1.1 Legal Constraints
1.2 Scope Alternatives

Correct return  change either direction

Deficiency assessment

Refund claim

Estimate effective use tax rate for a prospective compliance agreement
1. 3 Period and Entities To Be Examined

Records available

Statute of limitations

Problem areas identified on past audits
1.4 Budget for Government and Taxpayer

Cost of planning, data collection, evaluation, appeals

Time available
1.5 Sampling Risk

Risk that auditorsâ€™ estimate of total dollars of incorrect tax based on
the auditorsâ€™ sample is different than the true amount in the population.

Attempt to design and execute sampling plan to minimize sampling risk.
1.6 Tradeoff Cost and Sampling Risk

Cost of data collection and analysis versus

Cost of sampling risk ("cost of incorrect projection")
2.0 Sampling Methods

Method choice depends on knowledge of available methods

Acceptability to government and taxpayers
2.1 Sampling Method Categories

Detail Census (section 2.2)

Judgment and Block Sampling (section 2.3)

Statistical Sampling (section 2.4)
2.2 Detail Census

Examine each invoice in a population or a subpopulation.

No sampling risk.

High cost of data collection.
2.3 Judgment and Block Sampling
2.3.1 Characteristics of Judgment Sampling

Relies on auditorsâ€™ judgment about representativeness of sample.

Unlike statistical sampling, quantification of sampling risk is not possible
with judgment sampling.

Block sampling is a form of judgment sampling

Blocks often determined by how records are stored (day, week, month, invoice
number sequence, etc.)
2.3.2 Comments on Block Sampling from Dr. Edward Bryant

Dr. Edward Bryant was a consulting statistician to Pennsylvania Department
of Revenue and many nontaxrelated government agencies.

Block sampling provides no assurance that a month which has an average
volume of business will have an average number of errors.

I am reminded of the case in which two things were being compared:

Each had four legs. Each was 20 inches high. Each was brown. Each weighed
14 pounds. Neither had a medical record.

But one was a dog; and the other, a table.

It is my impression that the courts are requiring a measure of precision;
and block sampling cannot provide such a measure.
2.4 Statistical Sampling

Plan: specifies probability that items are selected for the sample.

Execution: requires a random number process to select sample items.

Evaluation: sampling risk is quantified based on statistical theory.
2.4.1 Simple Random Sampling

Population is not divided into stratum.

Each item in population has equal probability of being selected.

Reasonable method if expect all items have equal probability of error.
2.4.2 Stratified Random Sampling

Population is divided into strata according to known attributes.

Goal is to design strata so that error rate is homogeneous (similar) within
a stratum and heterogeneous (different) between strata.

Texas Comptroller Audit Division uses stratified random sampling on large
computerized audits, but does not perform the analysis of sampling risk
required for true statistical sampling.
3.0 Planning a Statistical Sample
3.1 Defining Population

Invoices with taxes accrued (possibly overpaid)

Invoices with no taxes accrued (possibly underpaid)

Selected divisions and time periods.
3.2 Sampling Unit

Invoice or line items

Credit invoices included in audit of purchases

Refunds included in audit of sales

Voided transactions

Date of invoice or delivery
3.3 Testing Sampling Frame

Sampling frame is the computer file of items that could be sampled, such
as accounts payable ledger or sales register.

Test that sampling frame is complete and accurate by comparing master list
and supporting records.

Estimate cost and time to retrieve supporting documentation.
3.4 Tolerable Error

Specify the maximum difference in tax dollars between the sample projection
and the true population value that would be tolerated (acceptable).

Fixed example: within $5,000 of population mean

Relative example: within 5% of population mean

Decreasing tolerable error increases sample size (and data costs)

Could depend on government policy, auditor judgment, and negotiation with
taxpayer.
3.5 Acceptable Sampling Risk

Specify the maximum acceptable probability that the sample projection differs
by more than the tolerable error from true population value

Example: 5% risk that projection and true value differ by more than $5,000
(also known as 95% confidence that projection is within $5,000 of true
amount in the population).

Decreasing acceptable risk increases sample size.

By stratum or population as a whole?
3.6 Stratification Attributes

What attributes are associated with changes in probability of error?

Invoice size is most common stratification attribute, but often no reliable
evidence that probability of tax errors systematically changes with invoice
size.

Consider detail census on strata with unusually high error rate or large
dollar invoices.
3.6.1 Possible Stratification Attributes

Invoice size category

Operating division

Slow versus busy months

Experience of clerks

Product line

Customer

Vendor

Delivery location

Areas effected by law change

etc.
3.7 Sample Leverage

Sample leverage is the amount the projected total error will change as
the result of an error in one sample item.

Sample leverage increases as:

number of items in stratum increases

average invoice amount increases
3.7.1 See "Sample Leverage Example"
3.8 Initial Sample Size (N)

Sample size increases as

tolerable error decreases

confidence requirement increases (or acceptable risk decreases)

population size increases

expected variance increases

Allocate sample size among strata

Consider expected variance and cost of data collection
3.9 Possible Procedures for Missing Documentation

Ignore and select replacement.

Treat all sample items with missing documentation as errors and include
in projection.

Estimate error rate on missing invoices at the average error rate on complete
invoices in the sample.

Detail missing invoices (assess tax only on sampled invoices with missing
documentation).

Compare missing invoice sample's observable characteristics (invoice size,
invoice date, operating department, etc.) to the characteristics of the
complete invoice sample.
3.10 Reviewing the Plan

Government's statistical experts

Taxpayer and advisers

Attempt to reach consensus on procedures before data collection.

Document areas where disagree on procedures.
4.0 Executing a Sampling Plan
4.1 Random Selection

Random numbers from printed tables or computer generated random number
function.

Sampling plan may specify higher probability of selection for some strata
than others.
4.2 Correctly Evaluate Each Sample Item

Validity of taxable or nontaxable determination.

Accuracy of tax rate by use, jurisdiction, and time period.

Review questionable items.
4.3 Follow Sampling Plan

Follow sampling plan's procedures for missing documentation, voids, credits,
etc.

Note issues that should be addressed in next audit plan for this taxpayer.
5.0 Evaluating Sample Results
5.1 Compare Results to Sampling Plan

Is there a reasonably small probability that the total amount projected
from the sample is not significantly different from the true amount in
the population?

Should the sampling plan be revised?

What additional data, if any, should be collected?
5.2 Example of Tolerable Error Goal

Goal: 95% confidence that point estimate of incorrect tax dollars is within
$5,000 of the true amount of incorrect tax dollars in the population.

Assuming a standard normal distribution ("bell curve"), 95% confidence
interval around mean = estimated mean +/ 1.96 standard errors.

Standard error around estimated mean equals standard deviation divided
by square root of sample size.

1.96 times estimated standard error < $5,000

Implies estimated standard error < $2,551
5.2.1 Example of Tolerable Error Achieved

Estimated mean = $100,000

Estimated standard deviation = square root of the sum of the squared differences
from the sample mean.

Estimated standard deviation = $20,000

Sample size (N) = 100

Square root of N = 10

Standard error = $20,000 / 10 = $2,000

(1.96) x $2,000 = $3,920

Estimated 95% C. I. = $100,000 +/ (1.96)($2,000)

Estimated 95% C. I. = ($96,080 , $103,920 )

Compare to tolerable error: $3,920 < $5,000

ACCEPT. The $100,000 estimate is within a 95% confidence interval
of the true mean.
5.2.2 Example of Tolerable Error Failure

Estimated mean = $100,000

Estimated standard deviation = square root of the sum of the squared differences
from the sample mean.

Estimated standard deviation = $40,000

Sample size (N) = 100

Square root of N = 10

Standard error = $40,000 / 10 = $4,000

(1.96) x $4,000 = $7,840

Estimated 95% C. I. = $100,000 +/ (1.96)($4,000)

Estimated 95% C. I. = ( $92,160 , $107,840 )

Compare to tolerable error: $7,840 > $5,000

FAILED. The $100,000 estimate is not within a 95% confidence
interval of the true mean.
5.3. Why Tolerable Error Not Achieved?

Sample size is too small.

Outliers (a few unusual items influencing estimates of mean and standard
deviation).

Data distribution is so skewed it is not close to the standard normal distribution.

Achieving the tolerable error goal does not eliminate nonsampling risk.
For example, auditors may have incorrectly determined the taxability of
a particular invoice in the sample.
5.4. Possible Actions To Revise Sampling Plan

Collect more data.

Remove outliers.

Change strata boundaries.

Revise tolerable error goal.

Project to lower bound of confidence interval

Other valid statistical procedures
5.5 Involvement of Taxpayer

Taxpayer's representatives and consultants should be involved in the sample
evaluation phase and discussions with examiner.

Collecting additional data and delaying final determination is costly to
the taxpayer.
6.0 Projecting Results
6.1 Minimum Number of Errors Required Before Projecting a Stratum

When very few errors exist in a stratum, the estimated variance and mean
have a higher risk they do not represent the true variance and mean in
the population.

California State Board of Equalization (SBE) Audit Manual, chapter 13,
"Statistical Sampling", section 1308.10 requires at least 3 errors
in a strata before projecting error from that sample.
6.2 Mean Projection

Also known as meanperunit estimator

Mean error in sample = ($ error in sample) / (number items in sample)

Projected error in population = (mean error in sample) x (number items
in population)

Could be sensitive to outliers, such as one extremely large error.
6.3 Ratio Projection

Sample error rate = (error $ in sample) / (total $ in sample)

Projected total error $ = (sample error rate) x (total $ in population)

Ratio projection tends to overstate total error when sample size is small.

Ratio projection is biased when data distribution is skewed, such as lots
of small invoices and relatively large invoices.
6.4 Point and Interval Estimates

Statistical practice requires both a point estimate (total $ of error)
and a confidence interval estimate (quantified estimate of the range around
the point estimate).

Wider confidence intervals indicate greater sampling risk.
6.5 Lower Bound of Confidence Interval

If achieved confidence intervals are wider than planned, consider adjusting
the estimate of total error, such as the lower bound of 95% confidence
interval.

Example:

Predetermined tolerable error = $5,000

Estimated mean = $100,000

95% CI = ( $92,160 , $107,840 )

Lower bound assessment at $92,160
6.6 Possible Appeals

Taxpayer and government consult statistical experts while preparing appeals.

Watch deadlines for filing appeals

Educating judges and hearing officers about the reliability of statistical
evidence.

Precedents and revised policies
6.7 Use Results for Planning Future Audits

Opportunities to improve compliance

Estimate error rates across invoice type

Identify factors associated with errors

Identify possible stratification variables

Opportunities to improve audit data collection process

Opportunities to improve tax compliance function
Bibliography at http://www.willyancey.com/sampling.htm
I welcome your comments suggestions for improving this document. Please
email any comments or suggestions to wyancey@aclrsbs.com