How good is he really?
The new signing has scored four penalties from four. Is he a 100% penalty taker? Nobody thinks so. But how good is he likely to be? Start from what you believed before, add his record, and see where the evidence leaves you.
Before you see him
His record
"How sure" is how many penalties' worth of evidence your belief is worth. Scored can't be more than taken.
Try an example
His record:
- Belief before
- Best estimate now
90% likely to be between .
Taller means more likely. A narrower curve means we're more certain.
How it works
The belief before is a Beta distribution: think of it as \(\alpha\) imaginary goals and \(\beta\) imaginary misses. His record adds real ones:
$$\begin{aligned} &\text{Beta}(\alpha, \beta) \\ &\longrightarrow\; \text{Beta}(\alpha + \text{scored},\; \beta + \text{missed}) \end{aligned}$$ $$\text{best estimate} = \frac{\alpha + \text{scored}}{\alpha + \beta + \text{taken}}$$In plain football
- Believing "a typical taker scores 75%, and I'm about as sure as 20 penalties would make me" is the same as starting with 15 imaginary goals and 5 imaginary misses.
- His 4 from 4 adds 4 real goals: now 19 goals from 24. Best estimate: 19 รท 24, about 79%.
- Each real penalty counts the same as each imaginary one, so the more he takes, the less your starting belief matters.
Small samples nudge; they don't decide
Four from four and a perfect record sound decisive. They aren't: an ordinary 75% taker scores four in a row about a third of the time, as the Binomial distribution article shows. So the estimate moves up, to about 79%, and the curve narrows a little, but most of what we know still comes from what penalty takers are like in general. By 45 from 50 the record does most of the talking.
This is Bayesian thinking: start with a sensible belief, let evidence move it, and let it move it more as the evidence grows. It is the same reason a manager doesn't rebuild a team after one good week.
Where it goes wrong
- The answer depends on the belief you start with. Two people with different starting beliefs will disagree until the evidence is large enough to overrule both.
- It assumes his true rate is fixed. A taker who has changed his technique really is a different player.
- A penalty is not only the taker: the goalkeeper and the pressure matter too.
These figures come from a statistical model and are for analysis and education. Football remains uncertain and model predictions will frequently be wrong.
The full explanation: Four from four. How good is he really? Bayesian thinking