Maths IA · Statistics and probability · topic 39 of 97
Goals and tries: are they random? · IB Maths IA idea: guide, sample graph and step-by-step plan
same mean, same variance: that's PoissonFootball goals and rugby tries seem to arrive without warning. If they really do, a Poisson model should predict how many 0–0 draws and 4–3 thrillers a season holds. Does it?
- ✓ you follow a sport and want to test a claim with numbers
- ✓ you like a model that has a single parameter to estimate
- ✓ you are happy to find out that a model only half fits
The big idea
If scoring chances are rare and independent, the number of goals (or tries) a team scores in a match follows a Poisson distribution, with one parameter λ, the average. You estimate λ from a season of results, predict how many matches have 0, 1, 2, 3 or more goals, and compare with what happened using a chi-squared test. The variance of a Poisson variable equals its mean, so the ratio of variance to mean is a second, simpler check. Real data can fail the test for reasons worth finding: strong and weak teams, home advantage, a team that goes ahead and defends. A model that does not fit is not a failed IA; explaining the gap is where the evaluation marks come from.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Estimate λ for all teams in a season, find the Poisson probabilities for 0, 1, 2... goals, and turn them into expected numbers of matches.
→ A set of expected counts to set against the real ones. Check they add up to the number of team-matches.Run a chi-squared goodness-of-fit test, pooling classes until each expected count is at least 5, and say how many degrees of freedom remain.
→ Showing the pooling and the degrees of freedom, including the one lost by estimating λ, is the core of the method.Estimate a separate λ for home and away teams and test whether the two rates differ.
→ Two models instead of one, and a question about whether the difference is more than chance.Compute variance divided by mean for your counts and decide whether it is close enough to 1.
→ A value above 1 means more variety than Poisson allows. Say what could cause it, such as teams of different strength.Repeat the analysis for goals in one sport and tries or points in another.
→ A different λ and possibly a different fit. Explain which sport is closer to Poisson and why.Give each team an attack and a defence rating and let λ depend on the opponent.
→ A model with many parameters. Compare its fit with the single-λ model and judge whether the extra work was worth it.Your Maths IA research question: run the check
which league? which season?Write your own question first. Then make sure it ticks every box:
- Names the competition, the season and what a data point is (one team in one match).
- States the model being tested and how you will decide whether it fits, with a significance level.
- Says how you will handle classes with small expected counts.
- Does not assume a good fit: you can report that the model fails and explain why.
- Says where the results come from, and that you checked a few entries by hand.
The model
One parameter λ gives every probability. Estimate it from your season, then test the shape. Draw the probabilities for your own λ.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
The competition and season, the claim you are testing, and what you expect before analysing: random scoring or not?
When it should apply, the formula, and why the mean equals the variance. Every symbol defined.
Where the results came from, how you cleaned them, and your estimate of λ for all teams and for home and away.
A table of observed and expected counts, with the pooling shown, and a bar chart.
Hypotheses, test statistic, degrees of freedom, p-value, and a conclusion in context.
Variance to mean ratio, home against away, and any extra model.
Back to your aim: whether scoring looks Poisson, and in what way it does not.
Independence of scoring chances, team strength, game state, small samples. Say which limitation matters most.
Where every data set and result came from.
What your Maths IA graph could look like
test the gap!Made-up numbers to show the form of a strong graph, not the answer: observed against expected counts of matches in each goal class. Points near the line mean a good fit. example only
- Observed and expected on the same axes, so any mismatch is visible at a glance.
- Axis labels say what is counted: team-matches in each goal class.
- Calculate the test statistic, then say what it means. A good-looking graph is not a test.
- Show the pooled classes in your table and explain how you chose them.
- Another graph to consider: variance to mean ratio for each team or each season.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out when counts follow a Poisson distribution. Before any calculation, guess whether goals will look Poisson and what will go wrong.
- Choose the dataPick one competition and season with results published by a source you can cite. State what one data point is.
- Clean the dataPut the goals of each team in each match in one column. Check three results by hand against the original source.
- Trial run (1 lesson)Estimate λ, work out the expected counts for 0 to 5 goals, and compare them with the real counts by eye. Fix any mismatch in the data now.
- Run the testPool classes so each expected count is at least 5, compute the chi-squared statistic, find the degrees of freedom and decide at a stated level.
- Check the dispersionCompute variance divided by mean. Decide how far from 1 is too far, and give a reason, such as a simulation.
- Split the dataRepeat for home and away, or for strong and weak teams, and compare what changes.
- Go further, with a reasonGive teams attack and defence ratings, or simulate a season from your model. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it says about scoring and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Poisson distribution – Use with estimated by the sample mean to find expected counts for each class.
- Expected value and variance – Show and use the ratio of sample variance to sample mean as a check.
- Chi-squared goodness of fit – Compute with pooled classes and degrees of freedom.
- Hypothesis testing – State and , a significance level and a conclusion in context; compare home and away means.
- Simulation – Simulate many Poisson seasons to see how far the variance to mean ratio varies by chance alone.
Key equations
Syllabus links
- 4.5Basic probability and expected frequency
- 4.7Discrete random variables and expected value
- 4.8Binomial distribution
- 4.3Averages and spread
- 4.5Basic probability and expected frequency
- 4.7Discrete random variables and expected value
- 4.8Binomial distribution
- 4.11Hypothesis tests: chi-squared and t-test
- 4.14Linear combinations of random variables; unbiased estimatesHL
- 4.17Poisson distributionHL
Read up first
- Probability distributions (AI HL) →
- Hypothesis testing (AI HL) →
- Univariate statistics (AI HL) →
- Using your GDC for statistics →
Data and sources to start from
Download one season of match results from a public results archive and record the goals or tries of each team in each match.
References to cite
- Maher, M. J. (1982). Modelling association football scores. Statistica Neerlandica, 36(3), 109–118.
- Dixon, M. J., & Coles, S. G. (1997). Modelling association football scores and inefficiencies in the football betting market. Journal of the Royal Statistical Society: Series C, 46(2), 265–280.
- football-data.co.uk. (2026). Football results, statistics & odds. https://www.football-data.co.uk/
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
count the degrees of freedom- Forgetting to subtract a degree of freedom for the estimated λ in the chi-squared test.
- Leaving classes with expected counts below 5 unpooled, which breaks the test.
- Treating a high p-value as proof that the model is true. It only says the data do not contradict it.
- Mixing strong and weak teams in one λ without saying so, then blaming 'randomness' for the misfit.
- Using a result with extra time or a walkover as an ordinary 90-minute match without a decision.
- Reading a variance to mean ratio of 1.1 as 'close to 1' with no idea of how much it varies by chance.
More Maths IA ideas like this
Waiting times: does a Poisson model fit?●●○
Hot streaks: is a player streaky or just luck?●●○
Tie-breaks: do league rules change who wins?●●○
Rating players: how does the Elo system work?●●○
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.