Goal or miss? The Bernoulli distribution
A penalty has two outcomes and nothing in between. That simple idea, the Bernoulli trial, is the building block of expected goals and of most football statistics.
Beginner Part 1
Statistics Through Football
Probability distributions from the ground up, from penalties and goals to streaks and transfer fees, one football question at a time.
12 parts Beginner to Intermediate
Start at part 1Five penalties, how many go in? The Binomial distribution
One penalty is a Bernoulli trial. Five penalties, each with the same chance, is the Binomial distribution, and it shows why missing one in five is normal, not a slump.
Beginner Part 2
How many goals will we score? The Poisson distribution
A team averages two goals a game. How likely is it to score exactly three? The Poisson distribution turns an average into a probability for every score, and it sits underneath most football prediction models.
Beginner Part 3
How long until the next goal? The Exponential distribution
Poisson counts goals. The Exponential distribution times the wait between them, and shows why "we're due a goal" isn't how probability works.
Beginner Part 4
How many shots until he scores? The Geometric distribution
A striker scores with one shot in five. How many shots until his first goal? The Geometric distribution answers it, and shows why a three-match drought is often just bad luck.
Beginner Part 5
How many shots until his hat-trick? The Negative Binomial distribution
The Geometric distribution waits for the first goal. The Negative Binomial waits for the third, or the fifth. It shows how long a hat-trick really takes, and it has a second job modelling goals that vary more than Poisson allows.
Intermediate Part 6
Win, draw or lose? The Multinomial distribution
A match has three possible results, not two. The Multinomial distribution handles any number of outcomes, and shows why a team's "expected" record over ten games almost never happens exactly.
Beginner Part 7
Eight from ten. Is he an 80% penalty taker? The Beta distribution
A striker scores 8 of his 10 penalties. Calling him an 80% taker is overconfident. The Beta distribution describes how sure we can really be about a probability, and how that changes as the evidence grows.
Intermediate Part 8
Will the third goal come before full time? The Gamma distribution
The Exponential distribution waits for one goal. The Gamma waits for several. It shows why a team that averages two goals a game gets its third before full time only about one match in three.
Intermediate Part 9
How fast is fast? The Normal distribution
Ranking players from fastest to slowest tells you the order, not how unusual anyone is. The Normal distribution, and its standard deviation, measures how far a player stands out from the rest.
Beginner Part 10
Is a goal equally likely in any minute? The Uniform distribution
The Uniform distribution says every outcome is equally likely. It's the fairest-sounding model in statistics, and testing it against 28,016 goals shows football isn't that fair.
Beginner Part 11
Why the average transfer fee misleads. The Log-Normal distribution
Transfer fees, wages and market values can't go below zero, cluster low and have a few enormous outliers. The Log-Normal distribution describes them, and shows why the average fee is a poor guide to a typical one.
Intermediate Part 12