Rounded data: calculate the interval behind a reported total
Rounded components do not always add to a separately rounded total. Before treating a one-unit discrepancy as a missing item or an extraction error, calculate the original-value intervals compatible with the displayed precision. The result can establish whether rounding is sufficient to explain the gap, without pretending to recover the hidden exact numbers.
The NIST rounding guidance discusses rounding and tie handling. It is not a universal issuer-reporting rule. Read the actual table’s precision and rounding convention; the interval worksheet below assumes rounding to the nearest stated unit unless another method is disclosed.
Convert displayed values into ranges
If a value is rounded to the nearest unit u, its original value is within half a unit of the display, subject to the tie convention. A conservative closed envelope is [display − u/2, display + u/2]. Exact endpoint inclusion depends on whether half cases round to even, away from zero or under another rule.
For addition, sum lower endpoints and upper endpoints separately. Compare that component-sum interval with the interval implied by the displayed total. An overlap means compatible underlying numbers may exist; it does not prove rounding is the actual explanation. A disjoint result means the stated rounding assumptions alone cannot reconcile the table.
Keep the units consistent. A number shown in millions to one decimal has a different half-unit range from an integer number of dollars. Do not give every table cell the same tolerance merely because the report uses one currency.
Hypothetical one-unit discrepancy
Assume a fictional table reports components of 12, 15 and 18 million, and a total of 46 million, all rounded to the nearest million.
| Displayed item | Conservative original-value envelope, millions |
|---|---|
| 12 | 11.5 to 12.5 |
| 15 | 14.5 to 15.5 |
| 18 | 17.5 to 18.5 |
| Component sum | 43.5 to 46.5 |
| Total of 46 | 45.5 to 46.5 |
The intervals overlap. For example, hypothetical originals of 12.4, 15.4 and 18.4 sum to 46.2. The components display as 12, 15 and 18, while the total displays as 46. No omitted fourth component is required by this example.
If the displayed total were 48, its envelope would be 47.5 to 48.5, disjoint from the component-sum envelope. Rounding alone would not explain that discrepancy under the stated assumptions. Investigate scope, eliminations, units, extraction and period alignment rather than enlarging the tolerance until the result passes.
Preserve uncertainty through derived metrics
For a difference, subtract the second interval’s upper endpoint from the first interval’s lower endpoint for a conservative lower limit, and reverse that pairing for the upper limit. A tiny displayed increase may therefore be compatible with no increase. Label this rounding uncertainty separately from estimation error or economic volatility.
For ratios, first inspect the denominator interval. If it crosses zero, ordinary endpoint division can fail to give a finite useful range. Do not publish a precise percentage change based on a denominator that could be zero after accounting for display precision.
Keep original displays, units, assumed convention, interval calculations and unresolved reconciliation items. Avoid rounding each intermediate step again; retain enough precision to evaluate compatibility and round only the presentation. A model should explain the interval, not invent extra digits that appear to solve the difference.
Recheck when higher-precision data becomes available, the issuer clarifies rounding, or a table’s scope changes. A compatible interval is evidence of possibility, not evidence that the underlying values are uniformly distributed within that interval.
The worksheet uses feasible sets, not probabilistic error distributions. It does not assume independent rounding errors or that positive and negative errors cancel. If several table identities must hold simultaneously, pairwise interval overlaps may not prove that one common set of originals satisfies all identities. Preserve those cross-table constraints and test them jointly before claiming a complete reconciliation. A compatible three-component example is only a local arithmetic check, not verification of an entire report.
FAQ
Does interval overlap prove the report is correct?
No. It shows the values could reconcile under the assumptions. Other errors may still exist.
Can I recover exact numbers by forcing the total to match?
No. Many underlying combinations may fit. Preserve the feasible ranges rather than presenting one invented combination as reported fact.