Abstract
Treatment $A$ wins in every subgroup and loses once the groups are pooled. This reversal is usually told as a statistical curiosity. This paper writes down that three different things travel under the one name. NO NEW MATHEMATICAL THEOREM AND NO NEW STATISTICAL METHOD IS CLAIMED.
First, it is arithmetic, not statistics. From $a/b<c/d$ and $e/f<g/h$ it does not follow that $(a+e)/(b+f)<(c+g)/(d+h)$. The smallest non-degenerate counterexample has denominators summing to $19$: $1/3<2/5$ and $5/7<3/4$, yet $3/5>5/9$ (Section 2).
Second, the mechanism is a difference of weights. In a table constructed here, $A$ wins in both groups yet loses on pooling by $0.54$ to $0.76$ — with exactly $100$ cases in each arm, so only the mixture differs (Section 3).
Third, the fence is balanced allocation. Searching all $669124$ tables with equal group sizes in both arms produced not one reversal — unbalancing them brings back $1166$ (Section 4).
Fourth, and this is the core. One table carries two correct answers. Read the group as a confounder and the adjusted averages are $0.70$ against $0.60$, so $A$ wins; read it as a mediator and the pooled averages are $0.54$ against $0.76$, so $B$ wins — with not a single number changed (Section 5).
Fifth, what must move for the answer to move is an arrow, not a number, and an arrow is not in the table (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW STATISTICAL METHOD IS CLAIMED — Yule, Simpson, Blyth, and the rules for when adjustment is licensed by a causal diagram are all standard. The table was constructed for this paper and is not real data — no medical or social claim of any kind is made. On Paper 124: that paper's “aggregation” is the licence to replace distant sources by a few quantities in an $N$-body calculation, not the pooling of subgroup statistics — the same word for a different thing. On Paper 154: the pattern that two operations fail to commute is that paper's, and no novelty is claimed for it here. What is added is that here which order is correct is neither taste nor convention: it is settled by a causal fact the data does not contain. What is added is separating the arithmetic, the mechanism and the causal question, finding the smallest non-degenerate counterexample by exhaustive search, confirming by exhaustive search that balanced allocation makes reversal impossible, and putting two causal structures on one table and getting opposite answers in exact fractions.
1. Notation and position
Write $A$ and $B$ for two treatments and split the cases into two groups $G_1$ and $G_2$. The success rate is successes over attempts. All arithmetic is in exact fractions and all searches are exhaustive.
Position: Paper 164 wrote that “probability” has four addresses. This paper stands beside it and exhibits something the data does not settle — one table yielding two correct answers.
2. It is arithmetic, not statistics
Observation 2.1 For fractions, $a/b<c/d$ together with $e/f<g/h$ does not give $(a+e)/(b+f)<(c+g)/(d+h)$. No probability, no sample, no model is involved — it is a fact about how fractions add.
Verification 2.2: Restricting to fractions strictly between $0$ and $1$ and searching exhaustively, the counterexample with the smallest sum of denominators is this.
| side $A$ | side $B$ | direction | |
|---|---|---|---|
| group $G_1$ | $1/3$ | $2/5$ | $A<B$ |
| group $G_2$ | $5/7$ | $3/4$ | $A<B$ |
| pooled | $\mathbf{3/5}$ | $\mathbf{5/9}$ | $\mathbf{A>B}$ |
The denominators sum to $19$. (QED)
Observation 2.3 So saying that “statistics is counterintuitive” misses the mark here. What is being violated is an intuition about fractions, and statistics has not yet begun.
3. The mechanism: pooling is a weighted average with different weights
Verification 3.1: The following table is constructed here. It is not real data.
| group $G_1$ | group $G_2$ | pooled | |
|---|---|---|---|
| treatment $A$ | $9/10=9/10$ | $45/90=1/2$ | $54/100=\mathbf{27/50}$ |
| treatment $B$ | $72/90=4/5$ | $4/10=2/5$ | $76/100=\mathbf{19/25}$ |
| direction | $A>B$ | $A>B$ | $\mathbf{A<B}$ |
Each treatment has exactly $100$ cases, so nothing here turns on one arm being larger. (QED)
Verification 3.2: What differs is the mixture.
| share in $G_1$ | share in $G_2$ | |
|---|---|---|
| treatment $A$ | $\mathbf{1/10}$ | $\mathbf{9/10}$ |
| treatment $B$ | $\mathbf{9/10}$ | $\mathbf{1/10}$ |
(QED)
Observation 3.3 The pooled number is each arm's average over its own mixture. It answers “what happens to a case of the kind that ends up in this arm” — which is not the question being asked.
4. The fence is balanced allocation
Observation 4.1 If both arms have the same group sizes, pooling is the same weighted average on both sides. An average with the same weights preserves order, so no reversal is possible.
Verification 4.2: Letting group sizes run to $12$ and searching all balanced tables exhaustively.
| allocation | tables examined | reversals |
|---|---|---|
| balanced (same sizes in both arms) | $669124$ | $\mathbf{0}$ |
| unbalanced | $512120$ | $\mathbf{1166}$ |
(QED)
Observation 4.3 Not one case among the balanced tables. This is what fixing the group sizes buys — not making reversal unlikely but making it logically impossible.
Observation 4.4 But this fence can only be built for groups one knows about. For groups one does not know, neither balancing the allocation nor adjusting afterwards is available.
5. The core: one table, two correct answers
Observation 5.1 Take the table of Section 3 unchanged. Not a single number is altered. Across everyone, $G_1$ is half and $G_2$ is half.
Verification 5.2: First reading — the group is a confounder (it affects both which treatment a case receives and how it turns out). Then the effect is the average weighted by each group's share of the whole population.
| treatment | adjusted average | verdict |
|---|---|---|
| $A$ | $7/10=\mathbf{0.7000}$ | |
| $B$ | $3/5=\mathbf{0.6000}$ | $\mathbf{A}$ is better |
(QED)
Verification 5.3: Second reading — the group is a mediator (the treatment itself moves cases between groups, and that movement is part of how it works). Adjusting would subtract the treatment's own effect, so the pooled number is the effect.
| treatment | pooled average | verdict |
|---|---|---|
| $A$ | $27/50=\mathbf{0.5400}$ | |
| $B$ | $19/25=\mathbf{0.7600}$ | $\mathbf{B}$ is better |
(QED)
Observation 5.4 Opposite answers from the same numbers. The table is identical in both cases, so no amount of studying it decides which reading is right.
6. What must move is an arrow, not a number
Observation 6.1
| causal structure | what to compute | answer |
|---|---|---|
| group $\to$ treatment (confounder) | adjusted average | $A$ |
| treatment $\to$ group (mediator) | pooled average | $B$ |
| treatment $\to$ group $\leftarrow$ outcome (collider) | pooled, do not condition | $B$ |
Observation 6.2 The mediator and the collider share an answer for opposite reasons. With a mediator, adjusting removes a real effect; with a collider, adjusting creates an association that is not there. Three structures and only two computations.
Observation 6.3 The separator is the direction of an arrow, and an arrow is not an entry in the table. When Paper 95 separated convention from fact it wrote that a convention must be a number and not an arrangement. Here the other side is needed — what settles the answer is not a number but a structure outside the numbers.
7. The census
Observation 7.1
| layer | what it is about | settled? |
|---|---|---|
| arithmetic | how fractions add | settled (Ver. 2.2) |
| mechanism | a difference of weights | settled (Ver. 3.2) |
| causal | which one to compute | not settled by the table |
Observation 7.2 The pattern that two operations fail to commute belongs to Paper 154 (a limit needs its order). No novelty is claimed for it here. What is added is how the correct order gets decided — in 154 the question fixes the order, while here it is fixed by a causal fact that the data does not contain.
8. Conclusion
- The reversal is arithmetic (Ver. 2.2): smallest denominators sum to $19$, with $3/5>5/9$.
- The mechanism is a difference of weights (Ver. 3.2): $100$ cases in each arm, mixtures $1/10$ against $9/10$.
- Balanced allocation makes reversal impossible (Ver. 4.2): $0$ among $669124$.
- Unbalancing brings it back (Ver. 4.2): $1166$ cases.
- One table carries two correct answers (Ver. 5.2, 5.3): $0.70$ against $0.60$ says $A$; $0.54$ against $0.76$ says $B$.
- Three structures, two computations (Obs. 6.2).
References
- Yule, G. U. (1903). Notes on the Theory of Association of Attributes in Statistics. Biometrika, 2(2).
- Fisher, R. A. (1935). The Design of Experiments. Edinburgh: Oliver and Boyd.
- Simpson, E. H. (1951). The Interpretation of Interaction in Contingency Tables. Journal of the Royal Statistical Society, Series B, 13(2).
- Blyth, C. R. (1972). On Simpson's Paradox and the Sure-Thing Principle. Journal of the American Statistical Association, 67(338).
- Pearl, J. (1995). Causal Diagrams for Empirical Research. Biometrika, 82(4).
- Pearl, J. (2009). Causality: Models, Reasoning, and Inference. Second edition. Cambridge: Cambridge University Press.
- Yamagishi, Y. (2026). Convention and Fact [Paper 95]. Zenodo. 10.5281/zenodo.21758621
- Yamagishi, Y. (2026). Three Distinct Roots of Aggregation [Paper 124]. Zenodo. 10.5281/zenodo.22057728
- Yamagishi, Y. (2026). The KL Divergence Is Not a Distance, but Its Second Derivative Is a Metric [Paper 137]. Zenodo. 10.5281/zenodo.22081369
- Yamagishi, Y. (2026). A Limit Without Its Order Is Not a Quantity [Paper 154]. Zenodo. 10.5281/zenodo.22084647
- Yamagishi, Y. (2026). Probability Has Four Addresses [Paper 164]. Zenodo. 10.5281/zenodo.22092815
- Yamagishi, Y. (2026). The Fence around Least Squares Is Linearity, Not Normality [Paper 171]. Zenodo. 10.5281/zenodo.22094287
- Yamagishi, Y. (2026). “The Curse of Dimensionality” Is Not One Curse [Paper 182]. Zenodo. 10.5281/zenodo.22104170
Reference
Yamagishi, Y. (2026). One Table, Two Correct Answers [Paper 183]. YOI: https://www.yamagishiyuuki.com/yoi/183
DOI: 10.5281/zenodo.22104230
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 183. This paper is a preprint and has not been peer reviewed.