- Independent Poisson
- 51.4%
- Correction delta
- -0.91 pp
Free · transparent rho · cell-by-cell proof
Dixon–Coles football score probability calculator
Apply the low-score correction without a black box. Compare all four adjusted cells, the complete score matrix and every 1X2 probability with the same independent-Poisson baseline.
All four tau termsFeasible rho guardPoisson referenceNot betting advice
Your transparent scenario
Set two goal rates and one visible rho
Rho is not guessed behind the interface. Enter it directly, stay inside the scenario-specific positivity interval, and see exactly which four score cells change.
Strict feasible interval for these lambdas-0.5882 < ρ < 0.5348
The bounds keep τ positive for 0–0, 0–1, 1–0 and 1–1. A boundary value is rejected rather than silently clipped.
Browser-local, deterministic arithmeticNo account, upload, team lookup or hidden fitted parameter. The URL fragment used for sharing is not a separate indexable page.
Dixon–Coles versus Poisson
One correction, three visible result shifts
Each card shows the corrected 1X2 probability, its independent- Poisson reference and the signed percentage-point difference.
- Independent Poisson
- 24.0%
- Correction delta
- +1.82 pp
- Independent Poisson
- 24.6%
- Correction delta
- -0.91 pp
Active correction contractPDC(h,a) = τ(h,a; λH, λA, ρ) × PPois(h,a)
Current ρ = -0.080. All cells outside 0–0, 0–1, 1–0 and 1–1 retain τ = 1.
This is a parameter explorer, not a rho estimator.A real rho must be fitted using information available before the test match, then evaluated out of sample. The calculator cannot turn a chosen value into evidence of accuracy.
Formula football-dixon-coles-scenario/1.0.0
The only four adjusted cells
Read tau before reading the headline probability
Tau is the multiplicative correction. Values above one raise a score probability; values below one lower it. Every displayed probability remains tied to the same λH, λA and ρ.
- τ multiplier
- 1.1496
- Poisson
- 6.08%
- Dixon–Coles
- 6.99%
- Delta
- +0.91 pp
- τ multiplier
- 0.8640
- Poisson
- 6.69%
- Dixon–Coles
- 5.78%
- Delta
- -0.91 pp
- τ multiplier
- 0.9120
- Poisson
- 10.34%
- Dixon–Coles
- 9.43%
- Delta
- -0.91 pp
- τ multiplier
- 1.0800
- Poisson
- 11.37%
- Dixon–Coles
- 12.28%
- Delta
- +0.91 pp
Tail-complete matrix
Every Dixon–Coles score probability, with its delta
Raised vs PoissonUnchangedLowered vs Poisson
Rows are home goals and columns are away goals. The 7+ row and column retain the displayed tail mass. Each cell gives the corrected probability first and DC-minus-Poisson beneath it.
| H \ A | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7+ |
|---|---|---|---|---|---|---|---|---|
| 0 | 6.99%+0.91 pp | 5.78%-0.91 pp | 3.68%0.00 pp | 1.35%0.00 pp | 0.37%0.00 pp | 0.08%0.00 pp | 0.01%0.00 pp | 0.00%0.00 pp |
| 1 | 9.43%-0.91 pp | 12.28%+0.91 pp | 6.25%0.00 pp | 2.29%0.00 pp | 0.63%0.00 pp | 0.14%0.00 pp | 0.03%0.00 pp | 0.00%0.00 pp |
| 2 | 8.79%0.00 pp | 9.67%0.00 pp | 5.32%0.00 pp | 1.95%0.00 pp | 0.54%0.00 pp | 0.12%0.00 pp | 0.02%0.00 pp | 0.00%0.00 pp |
| 3 | 4.98%0.00 pp | 5.48%0.00 pp | 3.01%0.00 pp | 1.10%0.00 pp | 0.30%0.00 pp | 0.07%0.00 pp | 0.01%0.00 pp | 0.00%0.00 pp |
| 4 | 2.12%0.00 pp | 2.33%0.00 pp | 1.28%0.00 pp | 0.47%0.00 pp | 0.13%0.00 pp | 0.03%0.00 pp | 0.01%0.00 pp | 0.00%0.00 pp |
| 5 | 0.72%0.00 pp | 0.79%0.00 pp | 0.44%0.00 pp | 0.16%0.00 pp | 0.04%0.00 pp | 0.01%0.00 pp | 0.00%0.00 pp | 0.00%0.00 pp |
| 6 | 0.20%0.00 pp | 0.22%0.00 pp | 0.12%0.00 pp | 0.05%0.00 pp | 0.01%0.00 pp | 0.00%0.00 pp | 0.00%0.00 pp | 0.00%0.00 pp |
| 7+ | 0.06%0.00 pp | 0.07%0.00 pp | 0.04%0.00 pp | 0.01%0.00 pp | 0.00%0.00 pp | 0.00%0.00 pp | 0.00%0.00 pp | 0.00%0.00 pp |
- #11–112.28%
- #22–19.67%
- #31–09.43%
- #42–08.79%
- #50–06.99%
- #61–26.25%
- #70–15.78%
- #83–15.48%
- #11–111.37%
- #21–010.34%
- #32–19.67%
- #42–08.79%
- #50–16.69%
- #61–26.25%
- #70–06.08%
- #83–15.48%
Short answer
What does the Dixon–Coles correction do?
It multiplies four independent-Poisson score probabilities by a tau term controlled by rho: 0–0, 0–1, 1–0 and 1–1. Every other exact score keeps tau equal to one. Negative rho usually raises the two low-score draws and lowers 0–1 and 1–0 for the same goal rates; positive rho reverses that direction.
The calculator deliberately exposes both distributions. That makes the correction inspectable without implying that a hand-entered rho is an estimated model parameter or that one corrected scenario is a real match forecast.
Piecewise formula
How is Dixon–Coles tau calculated?
Low-score multipliersτ(0,0) = 1 − λHλAρ
τ(0,1) = 1 + λHρ
τ(1,0) = 1 + λAρ
τ(1,1) = 1 − ρ
τ(h,a) = 1 otherwise
τ(0,1) = 1 + λHρ
τ(1,0) = 1 + λAρ
τ(1,1) = 1 − ρ
τ(h,a) = 1 otherwise
PDC(h,a) = τ(h,a) × PPoisson(h,a).
The strict feasible interval is not one universal range. It depends on both lambdas and requires every active tau to remain positive. This implementation rejects an invalid or boundary rho instead of clipping it and presenting a different model than the one requested.
Parameter integrity
How should rho be chosen for a real football model?
Estimate rho on eligible historical training matches only, together with a clearly declared goal-rate model. Freeze that fit before the test match, then evaluate held-out probabilities with proper scores. Do not select rho because one visible score matrix looks attractive, copy a fixed value from another league, or refit using the result being evaluated.
- Use as-of data and chronological train/test folds.
- Record the lambda model, rho fit and parameter version together.
- Check positivity constraints before scoring every test row.
- Compare paired held-out results with uncertainty, not one example.
First-party evidence
Does Dixon–Coles always improve football predictions?
No. In our 1,520-match, 143-test-week Premier League benchmark, a training-only sequential correction changed held-out exact- score log loss by +0.00032761 nats per match (Dixon–Coles minus Poisson; lower is better). The paired 95% interval was [−0.00041314, +0.00107077], so the design did not distinguish either method as better.
That study held the fitted lambdas constant and isolated one sequential rho correction. It was not a full joint Dixon–Coles maximum-likelihood fit and does not prove that Poisson always wins. Its useful conclusion is narrower: treat rho as a fitted, testable parameter—not a universal accuracy upgrade.
Transparent mathematics is not performance evidence.
The interactive result proves what the stated formula returns for the stated inputs. Only a frozen out-of-sample record can support an accuracy claim.
Primary research trail
Read the model definition, then the test design
- Dixon & Coles (1997), Modelling association football scores and inefficiencies
- Maher (1982), Modelling association football scores
- Football Proof AI, Poisson versus Sequential Dixon–Coles EPL Score Probability Benchmark
This page is an editorial technical note with an interactive browser tool. It has not been externally peer reviewed. It does not claim an externally certified implementation, fitted live model or profitable betting strategy.
Canonical publication record
Abstract
A deterministic browser-local Dixon-Coles parameter explorer that applies a user-supplied rho to 0-0, 0-1, 1-0 and 1-1, enforces the scenario-specific tau-positivity interval, and compares the complete normalized score matrix and 1X2 distribution with independent Poisson.
- Author and publisher
- Football Proof AI
- Technical report
football-dixon-coles-scenario/1.0.0- Published
- Last modified
- Release status
- Current release
- Review status
- Editorial technical note; not externally peer reviewed
- Version history
football-dixon-coles-scenario/1.0.0: Initial public release.