Audit sampling: separate projected and observed errors
An error found in a sample and an error projected to a population are different evidence objects. The first was observed in specified items; the second depends on the sampling design and estimator. A useful audit-research table retains both, together with the population, separately examined items and remaining sampling risk.
This guide concerns PCAOB issuer audits and AS 2315 as displayed on October 25, 2026. Its page flags an amendment to paragraph .11 effective December 15, 2026. The example uses the then-applicable text, not the future amendment. It is a fictional substantive test of monetary overstatement, not a control-deviation test or a recommendation for a real audit sample size.
Define the population and the direction of risk
AS 2315 .16–.24 connect sample design to the objective, population, tolerable misstatement, selection and separately examined items. Start by deciding what error the procedure could detect.
Sampling recorded invoices may reveal that some recorded amounts are overstated. It cannot, by itself, find every liability omitted from those records. A completeness objective may require a different source population, such as subsequent payments. A large sample from the wrong population does not repair that mismatch.
Retain the population extraction date, currency, book-value total, item count and selection method. Reconcile the extraction to the relevant ledger before projecting results. Otherwise a statistically sound selection can still describe an incomplete or incorrectly filtered list.
Work two distinct strata
All monetary figures are hypothetical thousands of currency units. The recorded population totals 1,000. A large-item stratum totaling 400 is examined completely and contains an observed overstatement of 8. Those items are excluded from the sampling stratum.
The remaining stratum totals 600 and consists of 200 invoices, each with recorded value 3. Assume a simple random selection of 20 invoices, all equally eligible and completely tested. Their recorded value is 60 and observed overstatement is 3. Equal recorded values make the illustrative monetary ratio equivalent to expanding the mean error per selected invoice.
| Stratum | Recorded population | Work performed | Observed overstatement | Evaluation amount |
|---|---|---|---|---|
| Individually examined large items | 400 | 100% examination | 8 | Known 8 |
| Homogeneous invoice stratum | 600 | 20 of 200 random invoices, value 60 | 3 | Projected 30 |
| Combined | 1,000 | Different procedures retained | Known sample findings total 11 | Point estimate 38 |
Projection for the second stratum = 3 / 60 × 600 = 30.
Combined point estimate = known large-item error 8 + sample-stratum projection 30 = 38.
The 30 already includes the sample’s observed 3. Adding 3 again would produce 41 and double count it. Known observed errors across both procedures are 8 + 3 = 11; 38 is not a claim that 38 of specifically identified invoices have already been demonstrated to be wrong.
Do not project the full-check layer again
AS 2315 .21–.22 and .26 distinguish items examined completely from those sampled and address projection to the appropriate population. The 400 layer is not part of the 600 expansion base.
Applying the 3/60 ratio to the full 1,000 and then adding 8 would produce 58. That calculation treats the fully examined layer as though it had the same sampled result, despite its directly observed error of 8.
Nor should the 8/400 result be imposed on the residual stratum merely because those large items were easy to inspect. Their complete examination answers a question about that layer; it does not make their error pattern representative of the remaining invoices.
In an actual unequal-value population, a ratio estimator, mean-per-unit estimator or another design may be appropriate under different conditions. The arithmetic here does not authorize using 3/60 with any arbitrary random list. Preserve selection probabilities, monetary structure and the estimator’s assumptions before interpreting a projection.
A point estimate is not a risk allowance
Assume, solely for the teaching example, a tolerable misstatement of 40 for the combined tested balance. The estimate of 38 lies only 2 below it. That numerical comparison does not establish acceptable risk that actual misstatement exceeds 40.
Sampling risk remains because another permitted sample could have produced another result. A statistical upper bound would require the design, variability, confidence assumptions and relevant calculations. None is supplied here, so the worksheet does not invent one or declare a pass.
AS 2315 .26–.30 require evaluation with sampling risk, qualitative matters and other evidence. An intentional error, an unexpected pattern or a finding inconsistent with planning assumptions can require more than comparing two totals.
Keep tolerable misstatement, the point projection and any supported risk allowance in different fields. Do not rename the projection “maximum possible error.” It is also not automatically the journal adjustment management should book; evaluation and accounting correction require identification of the affected facts and applicable requirements.
Mechanism-changing boundaries
Replace the random 20-invoice selection with a convenience selection of documents management supplied promptly. The recorded sample value can still be 60 and errors can still be 3, but the earlier representative-design assumption no longer holds. The same 30 arithmetic is not a supported population estimate.
Similarly, selecting only suspicious invoices can be valuable targeted work without producing a representative projection. Record directly found errors and investigate their implications. Do not claim that targeted selection is statistically equivalent because its sample fraction matches 10%.
In a different case, no errors are found among 20 properly selected invoices. The point estimate for the stated estimator may be zero, but sampling risk and unobserved population error do not disappear. Zero observed error is not proof that every invoice is correct.
Finally, if three selected invoices lack usable evidence, do not simply replace them or divide the remaining errors by the convenient tested subset as though all 20 were completed. Paragraph .25 addresses unexamined items, alternatives and implications for evaluation and risk. Retain their status and assess the consequences rather than silently treating them as error-free.
Execute a reproducible projection file
| Field | Required record | Why it matters |
|---|---|---|
| Objective | Assertion and error direction | Population can address the risk |
| Strata | Item lists and book totals | No overlap or missing layer |
| Selection | Design, counts and eligible frame | Representativeness can be assessed |
| Findings | Item-level measured differences | Known error distinct from inference |
| Projection | Estimator and exact expansion base | No duplicate sample or full-check error |
| Risk evaluation | Tolerable amount and supported uncertainty | Point estimate is not a pass threshold |
| Exceptions | Untested items and qualitative findings | Missingness and intent remain visible |
Private audit selections and complete invoice files are usually inaccessible to investors. Public filings may report a restatement, an audit limitation or a description of sampling, without disclosing the inputs needed to reproduce a projection. Do not derive an auditor’s actual tolerable amount or population error from a company’s public net income.
If totals conflict, first check gross versus net amounts, population filters, full-check exclusions and duplicate invoices. Then compare the error direction and estimator. Opposite-sign errors and different assertions should not be combined without explaining their relevance.
Update the file when the population changes, a selected item is resolved, a finding indicates a broader pattern, or corrected evidence changes the estimate. Preserve the original sample identifiers so later work can be distinguished from replacement selection or an expanded test.
Demonstrate why the estimator must be specified in advance
Consider a separate 200-invoice population totaling 600, now with unequal recorded amounts. A simple random sample of 20 happens to total 90 and contains measured overstatement of 3. Expanding the sample’s mean error per invoice gives 3/20 × 200 = 30. A monetary ratio calculation gives 3/90 × 600 = 20.
The different results do not establish that the lower answer is better. Each estimator needs its own justification in light of the design and population. In particular, selecting a method after seeing which result lies below tolerable misstatement undermines the evaluation. The original example used equal amounts precisely to avoid hiding this choice.
Retain the design rationale, sampling unit, eligible population, selection record and intended estimator before testing. Keep invoice count and book-value coverage in separate columns. If the sampling unit is a monetary unit rather than an invoice, do not quietly apply an invoice-mean formula.
A reproducible file also records how cancelled, zero, negative and duplicate entries were treated. Those decisions can alter which assertions the population supports. A clean total alone cannot demonstrate that every relevant item had the intended opportunity for selection.
If the separate fully examined layer still has known error of 8, the alternative projections would lead to combined point estimates of 28 or 38. The difference is a consequence to explain through the supported method, not a reason to select whichever answer makes the test appear acceptable.
FAQ
Should the sample’s error of 3 be added to 38?
No. It is already included in the 30 projection for the sampled stratum.
Does 38 below a tolerable 40 mean the balance passes?
No. Sampling risk and qualitative evidence still require evaluation.