Abstract
One way to estimate how many errors remain in a manuscript is to have two people read it separately and use the overlap of what they found. What this paper shows is that the estimate is right only when the two readers’ misses are unrelated; if they miss the same errors it comes out too few, if they miss opposite ones too many, and which way it errs is set by the sign of the covariance of their misses. No new theorem or law is claimed.
First, the overlap gives the total——if A finds $20$, B finds $25$ and both find $10$, the total is estimated at $\mathbf{50.000000}$ and the errors not yet found at $\mathbf{15.000000}$; the small-sample correction gives $\mathbf{48.636364}$ (Section 2).
Second, if findability is uniform, the estimate is exactly right——with $100$ true errors the expected estimate is $\mathbf{100.000000}$, and the estimate of those not found, $\mathbf{30.000000}$, matches the true $30$ (Section 3).
Third, and this is the core. If both readers miss the same errors, the estimate comes out too few——when half the errors are found by each reader with probability $0.7$ and half with $0.1$, the overlap swells to $\mathbf{25.0000}$, the total is estimated at $\mathbf{64.000000}$, and the $45$ errors not yet found are reported as $\mathbf{9.000000}$ (Section 4).
Fourth, if their blind spots are opposite, the estimate comes out too many——the overlap shrinks to $\mathbf{7.0000}$, the total is estimated at $\mathbf{228.571429}$, and $27$ errors not yet found are reported as $\mathbf{155.571429}$ (Section 5).
Fifth, the error of the estimate has one formula——the ratio of estimate to truth is $1/(1+\mathrm{cov}/(\bar p_1\bar p_2))$, matching the direct calculation to $10^{-12}$ (Section 6).
Sixth, the separator is whether the two readers’ misses are related, and in which direction——a positive covariance gives too few, a negative one too many, and only zero is right (Section 6).
Keywords: capture-recapture, Lincoln-Petersen estimator, proofreading, number of errors, independence assumption.
Scope (what is not claimed)
No new theorem or law is claimed——the Lincoln--Petersen estimate and the Chapman correction are standard in capture--recapture. Only expected values are computed——the variation from one trial to another and confidence intervals are not treated. Findability is split into two representative groups——half at $0.7$ and half at $0.1$; the distribution in real manuscripts is not stated. Only two readers are considered. No proofreading procedure is designed——who should read how many times is not stated. Relation to earlier papers: Paper 334 showed that combining independent measurements adds precisions, but that perfectly correlated sensors are one——here too the assumption of independence decides the number: there correlation broke the addition of precisions, here it breaks the estimated total, and by its direction the estimate errs either way. Paper 579 showed that a correct-response rate is not a property of the question alone but a relation between question and group——here too, that findability differs from error to error is what moves the number. Paper 300 showed that sharing a root is decidable——coming out too few and coming out too many share one root, the covariance of the misses. What is added is confirming by expected values that the estimate is exactly right when findability is uniform, setting out that shared misses report $45$ unfound errors as $\mathbf{9.000000}$ and opposite misses report $27$ as $\mathbf{155.571429}$, placing the error in a covariance formula checked against the direct calculation, and placing the separator on the sign of the covariance.
1. Notation and the Question
Setting: a manuscript with $N$ errors is read separately by A and B; A finds $n_1$, B finds $n_2$ and both find $m$. If their misses are unrelated, the fraction A found among all errors equals the fraction A found among those B found, so
$$\hat N=\frac{n_1\,n_2}{m}$$estimates the total (Lincoln--Petersen). The estimate of errors not yet found is $\hat N-(n_1+n_2-m)$.
Observation 1.1 errors not yet found seem uncountable, being unseen. Yet a large overlap between two readers suggests few errors, and a small one many——the same reasoning estimates the fish in a pond from how many marked and released fish reappear in a second catch.
Question: when is that estimate right, and when does it err, and which way?
2. Two Readers
Computation 2.1: A finds $20$ errors, B finds $25$, both find $10$. The total is estimated at $\mathbf{50.000000}$; $35$ were found, so $\mathbf{15.000000}$ are estimated not yet found. The small-sample correction (Chapman) gives $\mathbf{48.636364}$.
Observation 2.2 the number of the unseen comes from the overlap of the seen. If B also found half of what A found ($10/20$), B is taken to have found half of everything, and the whole is twice $25$, or $50$——this reasoning assumes that the two miss errors separately, without relation to each other.
3. If Findability Is Uniform
Computation 3.1: a true total of $100$, with every error found by A with probability $0.4$ and by B with $0.5$. In expectation $n_1=\mathbf{40.0000}$, $n_2=\mathbf{50.0000}$, overlap $\mathbf{20.0000}$, and the estimated total is $\mathbf{100.000000}$ (matching the true $100$ to $10^{-9}$).
Observation 3.2 if findability is uniform, the estimate is exactly right. $\mathbf{70.0000}$ are found, and the estimate of those not found, $\mathbf{30.000000}$, matches the true $30$——the two readers may differ in skill; as long as findability does not differ from error to error, the estimate does not err.
4. When Both Miss the Same Errors
Computation 4.1: of $100$ true errors, half are found by both readers with probability $0.7$ and half by both with only $0.1$.
| quantity | expected value |
|---|---|
| found by A | $\mathbf{40.0000}$ |
| found by B | $\mathbf{40.0000}$ |
| overlap | $\mathbf{25.0000}$ |
| estimated total | $\mathbf{64.000000}$ |
| found | $\mathbf{55.0000}$ |
| not yet found (estimated) | $\mathbf{9.000000}$ |
| not yet found (true) | $\mathbf{45.000000}$ |
Observation 4.2 this is the core. When both readers miss the same errors, the $45$ not yet found are reported as $9$——the easy errors are found by both and swell the overlap, while the hard ones are missed by both and appear neither in the overlap nor in the counts. A large overlap is misread as “almost all found”.
5. When They Miss Opposite Errors
Computation 5.1: the same two groups, but A finds the first half with $0.7$ and the second with $0.1$, while B does the reverse.
| quantity | expected value |
|---|---|
| overlap | $\mathbf{7.0000}$ |
| estimated total | $\mathbf{228.571429}$ |
| found | $\mathbf{73.0000}$ |
| not yet found (estimated) | $\mathbf{155.571429}$ |
| not yet found (true) | $\mathbf{27.000000}$ |
Observation 5.2 with opposite blind spots, the estimate comes out too many. Each reader covers the other’s misses, the overlap shrinks, and $27$ unfound errors are reported as $155.571429$——the same formula errs in either direction.
6. The Formula of the Error, and the Separator
Computation 6.1: from the mean findabilities $\bar p_1,\bar p_2$ and the covariance cov of findability across errors, the ratio of estimate to truth is
$$\frac{\hat N}{N}=\frac{1}{1+\mathrm{cov}/(\bar p_1\,\bar p_2)}.$$For the group that misses the same errors (covariance $0.0900$) and for the group that misses opposite ones (covariance $-0.0900$), it matches the direct calculation to $10^{-12}$.
Observation 6.2 the separator is the sign of the covariance of the two readers’ misses. A positive covariance (missing the same errors) gives too few, a negative one (missing opposite errors) too many, and only zero is right——whether the estimate is right cannot be told from the counts found; it is decided by an unseen property, whether the two are weak on the same kinds of error.
Observation 6.3 as in Paper 334, the assumption of independence decides the number. There, combining independent measurements added precisions, but perfectly correlated sensors were one——here correlation breaks the estimated total, and by its direction the estimate errs either way.
Observation 6.4 as in Paper 579, non-uniformity moves the number. There, questions of different discrimination reversed their order of correct-response rate between groups——here, errors of different findability, shared between the two readers, make the estimate come out too few.
Observation 6.5 one root, under Paper 300. Coming out too few and coming out too many come from one root, the covariance of the misses——only the sign differs.
Observation 6.6 what is not said. No proofreading procedure is designed——what is shown is only on what assumption an estimate from the overlap is right, and which way it errs when the assumption fails.
7. Roll Call and Closing
- $20$, $25$, overlap $10$: estimate $\mathbf{50.000000}$ (Computation 2.1): $\mathbf{15.000000}$ not yet found.
- small-sample correction $\mathbf{48.636364}$ (Computation 2.1): Chapman.
- uniform findability: estimate $\mathbf{100.000000}$ (Computation 3.1): exactly right.
- shared misses swell the overlap to $\mathbf{25.0000}$ (Computation 4.1): it swells.
- $45$ unfound reported as $\mathbf{9.000000}$ (Observation 4.2): the core.
- opposite misses shrink the overlap to $\mathbf{7.0000}$ (Computation 5.1): it shrinks.
- $27$ reported as $\mathbf{155.571429}$ (Observation 5.2): too many.
- the error is $1/(1+\mathrm{cov}/(\bar p_1\bar p_2))$ (Computation 6.1): matching the direct calculation.
- the separator is the sign of the covariance (Observation 6.2): positive too few, negative too many.
- independence decides, as in Paper 334 (Observation 6.3): correlation sets the number.
Closing: One way to estimate how many errors remain in a manuscript is to have two people read it separately and use the overlap of what they found. The number of the unseen comes from the overlap of the seen——if A finds $20$, B $25$ and both $10$, the total is estimated at $\mathbf{50.000000}$ and $\mathbf{15.000000}$ remain, or $\mathbf{48.636364}$ with the small-sample correction; with uniform findability the expected estimate hits the true $100$ exactly. But if both readers miss the same errors, the estimate comes out too few——with half the errors easy for both and half hard for both, the overlap swells to $\mathbf{25.0000}$, the total is estimated at $\mathbf{64.000000}$, and the $45$ not yet found are reported as $\mathbf{9.000000}$: a large overlap is misread as “almost all found”. With opposite blind spots the overlap shrinks to $\mathbf{7.0000}$ and $27$ unfound errors are reported as $\mathbf{155.571429}$. The separator is the sign of the covariance of the two readers’ misses——the ratio of estimate to truth is $1/(1+\mathrm{cov}/(\bar p_1\bar p_2))$: too few for a positive covariance, too many for a negative one, right only at zero. Whether the estimate is right cannot be told from the counts; it rests on an unseen property, whether the two are weak on the same errors. Under Paper 300 the two directions of error share one root. Placed among the earlier papers——as in Paper 334 independence decides the number, and here its direction makes the estimate err either way. As in Paper 579, non-uniformity moves the number. To be honest——only expected values were computed, with no variation or confidence intervals; findability was split into two representative groups and only two readers considered. And no proofreading procedure is designed: what is shown is on what assumption the estimate is right, and which way it errs.
References
- Chapman, D. G. (1951). Some Properties of the Hypergeometric Distribution with Applications to Zoological Sample Censuses. University of California Publications in Statistics.
- Lincoln, F. C. (1930). Calculating Waterfowl Abundance on the Basis of Banding Returns. United States Department of Agriculture Circular.
- Seber, G. A. F. (1982). The Estimation of Animal Abundance and Related Parameters (2nd ed.). Griffin.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). What Is Being Added Is Not the Values but the Precisions [Paper 334]. Zenodo. 10.5281/zenodo.22681804
- Yamagishi, Y. (2026). A Correct-Response Rate Is Not a Property of the Question but a Relation Between Question and Group [Paper 579]. Zenodo. 10.5281/zenodo.23014333
Reference
Yamagishi, Y. (2026). Errors Not Yet Found Can Be Counted from Two Readers’ Overlap, but Come Out Too Few If Both Miss the Same Ones [Paper 585]. YOI: https://www.yamagishiyuuki.com/yoi/585
DOI: 10.5281/zenodo.23014365
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 585. This paper is a preprint and has not been peer reviewed.