Recover Counts Compatible with Rounded Percentages

Preserve the published precision

When a report supplies a percentage and a denominator but omits the numerator, the percentage usually identifies a set of possible counts rather than a unique answer. Record the displayed digits exactly, including a trailing decimal place. A value printed as 12% contains different rounding information from one printed as 12.0%.

First verify that the denominator belongs to the same outcome and analysis population. Enrollment totals, available-case totals, weighted denominators, and percentages averaged across groups are not interchangeable. The reconstruction described here assumes an unweighted integer numerator divided by the stated integer denominator.

State the rounding rule

The Python decimal documentation distinguishes rounding modes, including rounding half upward for nonnegative values. The rounded percentage tool assumes that convention and zero to three decimal places. It is a declared assumption, not evidence of how the publication's software actually rounded.

If a percentage p is reported to d decimal places, its half-unit is 0.5×10^(−d) percentage points. Compatible unrounded percentages lie at or above p minus that amount and strictly below p plus that amount, with the feasible range bounded by zero and 100. The upper rounding boundary is excluded because a value exactly there rounds to the next displayed percentage. A true percentage of 100 remains possible when the displayed value is 100; clipping the feasible range does not exclude that endpoint.

Enumerate fictional possibilities

Suppose a fictional table reports 12% with denominator 250. The rounding interval is 11.5% through values below 12.5%. Multiplying by 250/100 gives a numerator interval from 28.75 through values below 31.25. Therefore counts 29, 30, and 31 are all compatible. Choosing 30 because it equals the displayed percentage times the denominator would hide genuine ambiguity.

If the same table reports 12.0% with denominator 250, the compatible percentage interval is 11.95% through values below 12.05%, producing count limits 29.875 and 30.125. Only 30 is compatible under the stated assumptions. Greater reported precision can narrow the set, but does not validate the denominator or rounding convention.

Handle empty and ambiguous results

No compatible count is a flag for investigation, not proof of a publication error. Recheck transcription, precision, weighting, denominator changes, and whether the percentage was truncated rather than rounded. Multiple compatible counts should remain a range or candidate set in an extraction note until another source resolves them.

Keep reconstructed counts separate from directly reported counts in the extraction dataset. Save the original text, page or table identifier, denominator source, rounding assumption, and compatible values. For a downstream analysis, examine whether the ambiguity changes the conclusion or request clarification from the source authors. A unique arithmetic candidate remains a conditional reconstruction; it does not become an independently verified observation merely because the calculator returns one integer.

Sources checked on 5 November 2026; all numerical examples are fictional and independently recomputed.