Eight from ten. Is he an 80% penalty taker? The Beta distribution

Contents

• The football question
• The concept
• A football example
• How sure we are, in one number
• More penalties, more certainty
• The same idea, all over the pitch
• Why it matters
• Limitations
• Try it yourself
• Further reading

The football question

A striker has taken 10 penalties and scored 8. The easy answer is that he's an 80% penalty taker.

But ten kicks is a small sample. Would you bet he's better than a 70% taker? Better than 90%? The honest answer is: he's probably somewhere around 80%, but we can't be sure yet. The Beta distribution puts numbers on that.

The concept

Everything in this series so far has taken the probability as known. The Binomial says: if he scores 80% of his penalties, here's how many he'll score from ten.

The Beta turns that around. It describes what we believe about the probability itself:

• Binomial: the probability is fixed, and the outcomes are uncertain.
• Beta: the outcomes are known, and the probability is uncertain.

It needs two numbers, written Beta(a, b). The simplest way to read them is as goals and misses: a counts the successes and b the failures, each plus one.

A football example

Start knowing nothing: every conversion rate from 0% to 100% is equally plausible. That's Beta(1, 1), a flat line.

Now add what he did: 8 goals, 2 misses.

$$\text{Beta}(1 + 8,\ 1 + 2) = \text{Beta}(9, 3)$$

Here's what Beta(9, 3) says his true conversion rate could be:

The single most likely value is 80%, his raw record. But there's a 31% chance he's actually below 70%, and the range that covers 90% of our belief runs all the way from 53% to 92%. Ten penalties tell us he's decent. They don't tell us much more than that.

How sure we are, in one number

The average of Beta(9, 3) is

$$\frac{a}{a + b} = \frac{9}{12} = 75\%$$

More penalties, more certainty

Suppose instead he'd scored 80 from 100: the same 80%, ten times the evidence. That gives Beta(81, 21).

Record	Best guess	90% likely between	Chance he's below 70%
8 from 10	75%	53% and 92%	31%
80 from 100	79%	73% and 86%	1.4%

Same conversion rate, very different confidence. That's the whole point of the Beta: it keeps the percentage and how much evidence sits behind it together, where a raw percentage throws the second part away.

The same idea, all over the pitch

• Penalty conversion: goals from penalties taken.
• Shot conversion: goals from shots.
• Tackle success: tackles won from tackles attempted.

Any time a player's record is "so many successes from so many tries", the Beta describes how well that record pins down his real ability.

Show the mathsThe formula, its mean and spread, and why the update is so simple. Optional.

The Beta distribution for a probability \(\theta\) between 0 and 1 is

$$f(\theta) = \frac{\theta^{a-1}(1-\theta)^{b-1}}{B(a, b)}$$

where \(B(a, b)\) is the constant that makes it add up to 1. Its mean and variance are

$$E[\theta] = \frac{a}{a+b}$$

$$\text{Var}(\theta) = \frac{ab}{(a+b)^2(a+b+1)}$$

For Beta(9, 3) that's a mean of 0.75 and a standard deviation of 0.12; for Beta(81, 21), 0.79 and 0.04. Its peak (the mode) is \((a-1)/(a+b-2)\), which is exactly 80% for both.

With a Beta prior and s successes from n Binomial trials, the result is \(\text{Beta}(a + s,\ b + n - s)\): the update is adding goals to one number and misses to the other. A prior that stays in the same family after the update is called conjugate.

Why it matters

Football is full of small samples dressed up as facts: a keeper who's saved three of his last four penalties, a striker converting 30% of his shots after six games. The Beta distribution is the discipline of asking how much that record really tells you. It's also the foundation of Bayesian thinking. Here we started from "no opinion", but you can start from what you already know about penalty takers in general, which is the subject of Bayesian Football.

Limitations

• The starting point matters with small samples. Beta(1, 1) says a 20% penalty taker is as plausible as an 80% one, which no football fan believes. Most takers score about three in four; starting there is fairer, and where priors come from shows how.
• It assumes one true rate. A taker's real rate can drift with confidence, age or a change of technique.
• Every attempt counts the same. A penalty against a specialist keeper in a cup final is treated like any other.

Try it yourself

Pick a player's penalty record. Add one to his goals and one to his misses, and divide goals by the total: that's the Beta's best guess. Then compare two players with the same percentage but very different numbers of penalties, and ask which one you'd really trust.

Further reading

• Beta distribution: properties, proofs, exercises, StatLect. The full derivations.
• Beta distribution, Wikipedia. The formal definition, and its role as a conjugate prior.