Why a Rounded Percentage May Not Reveal the Count
A report might say that 27% of a sample had a particular outcome. If the sample size is known, it is tempting to multiply \(0.27\) by that size and treat the result as the number of successes. But the reported percentage may have been rounded: several different integer counts can produce the same displayed value. This matters whenever an analysis, including a one-proportion test, needs the actual number of successes rather than just a rounded summary.
The task is to work backward from the reported percentage and sample size. Sometimes the information identifies exactly one count; sometimes it identifies a range of possible counts. In the latter case, calculations based on a single guessed count can give an unjustified answer.
Turning a Rounded Percentage into a Count Range
Suppose a percentage \(r\) is reported to \(d\) digits after the decimal point. The rounding step is \(h=10^{-d}\) percentage points: for example, a whole-number percentage has \(h=1\), while a percentage reported to one decimal place has \(h=0.1\). Under the common round-half-up rule for nonnegative values, a value at least \(r-h/2\) but less than \(r+h/2\) rounds to \(r\). Thus a compatible count satisfies
Multiplying by \(n/100\) gives an interval for the count:
The endpoints of this interval need not be integers. The compatible counts are the integers inside it, also restricted to \(0\leq x\leq n\). The lower endpoint is included and the upper endpoint excluded because of the round-half-up convention. If a possible percentage is exactly halfway between two displayed values, a different rounding rule—such as round-half-to-even—could assign it differently. When endpoint cases matter, check the reporting convention rather than silently assuming one.
Worked Examples: Recovering Compatible Counts
Worked Example: A Percentage That Identifies One Count
A report says that 18.3% of 480 respondents selected an option. Assume the percentage was rounded to the nearest tenth of a percentage point using round-half-up. How many successes are compatible with the report?
Here \(r=18.3\), \(h=0.1\), and \(n=480\). The count interval is
so
The only integer in this interval is \(x=88\). As a check, \(100(88/480)=18.333\ldots\%\), which rounds to 18.3%. In this case, the reported percentage and sample size determine the count uniquely.
Worked Example: Several Counts Produce the Same Whole Percentage
A sample contains 800 observations, and a report gives a success percentage of 27%, rounded to the nearest whole percentage point using round-half-up. Find all compatible success counts.
Now \(r=27\), \(h=1\), and \(n=800\). The interval for \(x\) is
which simplifies to
Therefore \(x\) could be any integer from 212 through 219. For instance, \(100(212/800)=26.5\%\), which rounds up to 27%, while \(100(219/800)=27.375\%\), which rounds to 27%. But \(x=220\) gives \(27.5\%\), which rounds to 28% under the stated convention, so it is not compatible. The report does not establish one exact count.
Why Count Ambiguity Can Affect a One-Proportion Test
A one-proportion test uses the observed count through \(\hat{p}=x/n\). For example, in the usual one-proportion z test, the null model is \(X\sim\operatorname{Binomial}(n,p_0)\): there are \(n\) independent trials, each with the same success probability \(p_0\) under \(H_0\). The test compares the observed sample proportion with \(p_0\), using the null standard error \(\sqrt{p_0(1-p_0)/n}\). Its statistic is
The normal approximation for this test requires the null expected success and failure counts, \(np_0\) and \(n(1-p_0)\), to be sufficiently large for the approximation to be reasonable. The test also relies on the independent, common-probability Bernoulli-trial model. A rounded report does not verify those conditions; they must be assessed for the underlying data and sampling design. When the actual count is uncertain, calculate \(\hat{p}\), \(z\), and the p-value for each compatible count rather than substituting the displayed percentage as if it were exact.
Worked Example: Plausible Counts Give Different Test Conclusions
Worked Example: Testing a Rounded 40% Report
A hypothetical report states that 40% of \(n=10{,}000\) observations were successes, rounded to the nearest whole percentage point using round-half-up. Consider a two-sided test of \(H_0:p=0.405\) against \(H_a:p\ne 0.405\), using a one-proportion z test at \(\alpha=0.05\). Show how the compatible counts affect the result.
First, find the counts compatible with the report:
Thus every integer count from 3950 through 4049 is possible. The test's null standard error is
Under \(H_0\), the expected success and failure counts are \(10{,}000(0.405)=4050\) and \(10{,}000(0.595)=5950\). Both are large, so the normal approximation is reasonable under the stated independent-trial model. The following calculations use the two-sided normal-approximation p-value \(2P(Z\geq |z|)\), where \(Z\sim N(0,1)\).
For the compatible count \(x=3950\), \(\hat{p}=3950/10{,}000=0.395\), giving
The two-sided p-value is approximately \(0.0417\), so this candidate count leads to rejection of \(H_0\) at the 0.05 level. For another compatible count, \(x=4000\), \(\hat{p}=0.400\), and
Its two-sided p-value is approximately \(0.309\), so this candidate does not lead to rejection. A compatible count near 4049 gives \(\hat{p}=0.4049\), \(z\approx-0.0204\), and a two-sided p-value of about \(0.984\), also not significant. The rounded report alone therefore does not determine the test conclusion: compatible counts can lead to different decisions. The correct response is to obtain the exact count or report that the conclusion cannot be settled from the rounded percentage alone.
A Short Justification for Checking Every Integer
The displayed percentage constrains \(100x/n\) to a rounding interval; it does not specify where inside that interval the unrounded percentage lies. Since \(x\) must be an integer, every integer in the corresponding count interval is a candidate. There is no general reason to prefer one candidate—such as the integer nearest \(nr/100\)—unless additional information about the unrounded data is available. And because the test statistic depends on \(x\), different candidates can produce different evidence against the same null value.
A careful analysis therefore separates two questions: Which counts could have produced the report? and What does each of those counts imply for the test? Answering only the first by multiplying the displayed percentage by \(n\) can miss compatible counts. Answering the second with an invented count can overstate what the published information supports.
Practical Checks and Common Mistakes
- Treating the displayed percentage as exact. A reported 27% usually means the underlying percentage lies in a rounding interval, not that it equals exactly 27.000%. Use the interval implied by the number of digits reported.
- Rounding the product \(nr/100\) and stopping. The nearest integer may be one compatible count, but it is not necessarily the only one. Find the full interval and enumerate its integers.
- Ignoring the rounding rule at a midpoint. A percentage exactly halfway between displayed values may be assigned differently under different conventions. State the assumed rule or check the source's rule if an endpoint is decisive.
- Assuming any published percentage came from a raw count divided by \(n\). The value might instead be weighted, adjusted, based on a different denominator, or calculated after excluding missing responses. Confirm what the percentage represents before attempting count recovery.
- Reporting a test decision from one convenient candidate. If compatible counts lead to different p-values or decisions, the rounded report does not support a single conclusion. Seek the exact count or report the range of possible results.
Check Your Understanding
Use round-half-up for nonnegative percentages unless a question says otherwise.
- A sample of 250 has a reported success percentage of 12.4%, rounded to the nearest tenth of a percentage point. Find the compatible integer counts.
- A sample of 600 is reported as 42%, rounded to the nearest whole percentage point. Write the interval for the count and list all compatible integers.
- Why is multiplying a rounded percentage by the sample size and rounding the result not always enough to recover the count?
- In a one-proportion test, explain why two counts compatible with the same report can give different test statistics.
- If all compatible counts lead to the same decision at a stated significance level, what uncertainty remains, and what should be reported about it?