Probability-proportional-to-size sampling in auditing, more often called monetary unit sampling, treats every dollar in a recorded balance as a sampling unit so that higher-value items are more likely to be selected. To use it, you set a tolerable misstatement, a confidence level, and an expected misstatement; divide tolerable misstatement by a reliability factor from a Poisson table to get a sampling interval; select items by walking that interval through a cumulative-dollar listing; and then compare an Upper Error Limit built from three components against your tolerable misstatement to reach a conclusion.
Set the Three Inputs First
Every calculation that follows depends on three numbers, and all three require judgment before you touch the population.
Tolerable Misstatement (TM). The largest dollar error you can accept in the account before concluding the financial statements are materially misstated. TM sits below overall materiality, commonly between 50% and 75% of it, with the exact percentage tied to the assessed risk for the account.1Public Company Accounting Oversight Board. Auditing Standard No. 11
Risk of Incorrect Acceptance (RIA), i.e., confidence level. The probability you are willing to accept of concluding a materially misstated balance is fairly stated. A 95% confidence level means a 5% RIA. Lower risk tolerance produces a larger sample.
Expected Misstatement (EM). Your best estimate of the dollar errors the sample will surface. Setting EM at zero produces the smallest sample. Prior-year errors or weak controls push EM higher and enlarge the sample.
The confidence level maps directly to a Reliability Factor from a standard Poisson table. That factor drives every subsequent formula.
The Reliability Factor Table
You cannot run PPS without these numbers. The factors below come from the Poisson distribution and feed the sample size, basic precision, and incremental allowance calculations.2European Commission. Guidance on Sampling Methods for Audit Authorities
Zero-error factors at common confidence levels:
- 99% confidence (1% RIA): 4.61
- 95% confidence (5% RIA): 3.00
- 90% confidence (10% RIA): 2.31
- 85% confidence (15% RIA): 1.90
- 80% confidence (20% RIA): 1.61
When errors turn up during evaluation, higher factors apply. At 95% confidence:
- 0 errors: 3.00
- 1 error: 4.74
- 2 errors: 6.30
- 3 errors: 7.75
The gap between consecutive factors matters later for the incremental allowance. From one error to two at 95% confidence, the gap is 1.56 (6.30 − 4.74). Keep the table at hand throughout the engagement.2European Commission. Guidance on Sampling Methods for Audit Authorities
Calculate the Sampling Interval and Sample Size
The sampling interval is the dollar gap between selection points. Divide tolerable misstatement by the zero-error reliability factor:
Sampling Interval = Tolerable Misstatement ÷ Reliability Factor
For a $5,000,000 accounts receivable balance with TM of $150,000 at 95% confidence, the interval is $150,000 ÷ 3.00 = $50,000.
Sample size follows from the interval:
Sample Size = Book Value ÷ Sampling Interval
$5,000,000 ÷ $50,000 = 100 items. Round up on any fraction; you cannot test half an item.
When EM is greater than zero, multiply it by an expansion factor (typically 1.6 at 95% confidence) and subtract that product from TM before dividing by the reliability factor. The interval shrinks, the sample grows, and you gain more evidence for the errors you expect. If EM is zero, use the basic formula.
Select the Sample Items
You need a complete cumulative-dollar listing of the population, reconciled to the general ledger. Each line item occupies a range of cumulative dollars. If customer A owes $12,000 and customer B owes $8,000, A spans dollars 1 through 12,000 and B spans 12,001 through 20,000.
Pick a random starting point between 1 and the sampling interval. With a $50,000 interval, that starting point might be dollar 23,417.3Statistics Canada. 3.2.2 Probability Sampling The transaction containing that dollar becomes item one. Add $50,000 for the next selection point (73,417), and continue until you reach the population total.
Items Larger Than the Interval
Any line whose book value equals or exceeds the sampling interval is guaranteed to be hit. These “top stratum” items are audited in full, and any misstatement found in them is used directly rather than projected.4Diligent One Platform Help. Performing Monetary Unit Sampling
Zero and Negative Balances
PPS cannot select items with zero or negative book values through normal systematic selection. Zero-balance receivables could hide fictitious write-offs; credit balances could mask understatement. Standard practice is to strip zero and negative items out of the PPS population and test them separately using judgmental selection, a threshold approach, or classical variables sampling.
Evaluate the Results
After auditing each selected item and documenting every misstatement, you calculate the Upper Error Limit (UEL). The UEL estimates the maximum possible overstatement in the account at your chosen confidence level. It has three parts.
Basic Precision
Basic Precision is the sampling risk that remains even with zero errors found. Multiply the sampling interval by the zero-error reliability factor:
Basic Precision = Sampling Interval × Reliability Factor (0 errors)
In the running example, $50,000 × 3.00 = $150,000. That equals TM by design: if no errors surface, UEL equals Basic Precision and the account passes.
Projected Misstatement and Tainting
How you project an error depends on whether the misstated item was above or below the sampling interval.
For top stratum items, record the actual dollar misstatement with no projection.
For items smaller than the interval, compute a tainting percentage: misstatement divided by recorded book value.5Diligent One Platform Help. Evaluating Errors in a Monetary Unit Sample Multiply that percentage by the sampling interval to project the error.
An invoice with a $4,000 book value overstated by $1,000 has a tainting of 25%. Projected misstatement is 25% × $50,000 = $12,500. A $1,000 error in one item represents $12,500 of estimated misstatement in the population, which is why small errors in PPS can have outsized effects.
Total Projected Misstatement sums the actual errors from top stratum items and the projected errors from smaller items.
Incremental Allowance
When two or more errors appear in the sample, add an Incremental Allowance to reflect the increased likelihood of further undetected errors. One error might be noise; several suggest a pattern.
Rank the projected misstatements from non-top-stratum items largest to smallest. For each error, use the gap between the reliability factor for that error number and the factor for one fewer, then subtract 1.00 because the projected misstatement is already captured in the PM component.2European Commission. Guidance on Sampling Methods for Audit Authorities
At 95% confidence: the incremental factor for the first error is 4.74 − 3.00 = 1.74, so the multiplier applied to that error is 0.74. For the second, 6.30 − 4.74 = 1.56, so the multiplier is 0.56.
With only one misstatement, the incremental allowance is zero. Basic Precision already covers the sampling risk of a single error.
Reach a Conclusion
Upper Error Limit = Basic Precision + Projected Misstatement + Incremental Allowance
Compare UEL to TM. If UEL is at or below TM, you can conclude the balance is not materially overstated at your chosen confidence level. If UEL exceeds TM, the sample does not support that conclusion. The account may be materially misstated.
When UEL exceeds TM, three options exist: expand the sample, perform alternative substantive procedures, or ask the client to investigate and adjust. Most engagements combine correction of known errors with targeted follow-up.
Look at Errors Qualitatively, Not Just by Amount
Standards require you to investigate the nature and cause of every misstatement, even when the UEL clears TM.6Public Company Accounting Oversight Board. Qualitative Factors Related to the Evaluation of the Materiality of Uncorrected Misstatements A $500 timing cutoff error reads very differently from a $500 error that reflects deliberate revenue inflation.
Look for concentration and direction. Three errors in one product line matter regardless of dollar amount. Errors that are always overstatements point to possible management bias. The PCAOB flags bias and the systematic accumulation of small errors as factors that can make quantitatively immaterial misstatements material in context.6Public Company Accounting Oversight Board. Qualitative Factors Related to the Evaluation of the Materiality of Uncorrected Misstatements Document the qualitative assessment alongside the numbers.
A Full Worked Example
Accounts receivable book value: $300,000 across 300 accounts. TM: $27,000. EM: zero. Confidence: 95%.
Reliability factor for zero errors at 95%: 3.00. Sampling interval: $27,000 ÷ 3.00 = $9,000. Sample size: $300,000 ÷ $9,000 = 33 items.
Random starting point: 4,215. Item one is the customer whose cumulative range contains dollar 4,215. Next selections are 13,215, 22,215, and so on until the population total.
Two overstatement errors turn up, both in items smaller than the $9,000 interval:
- Error A: book value $1,650, audited value $572.55. Misstatement $1,077.45. Tainting 65.3%. Projected misstatement 65.3% × $9,000 = $5,877.
- Error B: book value $975, audited value $159.90. Misstatement $815.10. Tainting 83.6%. Projected misstatement 83.6% × $9,000 = $7,524.
Evaluation:
- Basic Precision: $9,000 × 3.00 = $27,000.
- Total Projected Misstatement: $5,877 + $7,524 = $13,401.
- Incremental Allowance: rank projected errors largest first. Error B ($7,524) × (4.74 − 3.00 − 1.00) = $7,524 × 0.74 = $5,568. Error A ($5,877) × (6.30 − 4.74 − 1.00) = $5,877 × 0.56 = $3,291. Total: $8,859.
- Upper Error Limit: $27,000 + $13,401 + $8,859 = $49,260.
The $49,260 UEL far exceeds the $27,000 TM. The auditor cannot conclude the balance is fairly stated. Next step: ask the client to investigate the errors, correct what can be corrected, and then decide whether to expand the sample or apply alternative procedures.
When PPS Is Not the Right Tool
PPS suits most balance testing but has real limits worth naming so you do not misapply it.
Understatement risk. Selection is proportional to recorded book value, so PPS tests overstatement well and understatement poorly. Missing liabilities and understated assets have no book value to select on. Classical Variables Sampling handles both directions.
High expected error rates. PPS sample sizes are smallest when errors are rare. As EM climbs, PPS samples grow quickly and can exceed what classical sampling would need for the same population.
Many zero or negative balances. PPS cannot select these items through normal systematic selection. When they make up a significant share of the population, running a separate procedure alongside PPS adds complexity that classical sampling avoids by treating each physical item as a sampling unit.
Inflated UELs from many small errors. Because tainting projects a small error across the full interval, populations with many small errors can produce a UEL that blows past TM even when actual total misstatement is modest. Classical Variables Sampling uses means and standard deviations and often produces tighter bounds when errors are scattered.