How fast is fast? The Normal distribution

Contents

• The football question
• The concept
• A football example
• Unusual, but every squad has one
• Comparing different skills
• The same idea, all over the pitch
• Why it matters
• Limitations
• Try it yourself
• Further reading

The football question

A winger clocks a top speed of 34 km/h. The squad average is 30. Is he quick, or is he exceptional?

Ranking the squad from fastest to slowest gives you the order, but not the gap. To know how unusual 34 km/h really is, you need to know how spread out everyone else is.

The concept

The Normal distribution is the bell curve. Most values bunch around the middle, and fewer and fewer sit further out on either side, symmetrically. It needs two numbers:

• μ (mu), the mean: what's typical.
• σ (sigma), the standard deviation: how spread out the values are around it.

The mean tells you what's normal. The standard deviation tells you how unusual anything else is.

A football example

Say players' top sprint speeds have a mean of 30 km/h and a standard deviation of 2 km/h. How far above the average is our winger, in standard deviations?

$$z = \frac{x - \mu}{\sigma} = \frac{34 - 30}{2} = 2$$

Every Normal distribution follows the same rule of thumb:

• About 68% of players are within one standard deviation of the mean: 28 to 32 km/h.
• About 95% are within two: 26 to 34 km/h.
• About 99.7% are within three: 24 to 36 km/h.

So only about 2.3% of players reach 34 km/h, roughly one in 44. Here's the whole picture:

Unusual, but every squad has one

One player in 44 sounds rare. But a squad has about 25 players. Pick 25 at random from this distribution and there's a 44% chance at least one of them reaches 34 km/h. The fastest of the 25 averages about 33.9 km/h.

So a two-standard-deviation player is exceptional compared to the typical player, and roughly what you'd expect the quickest player in a squad to be. Whether he's special depends on which question you're asking.

Comparing different skills

The z-score's real strength is that it has no units. Say distance covered in a match averages 10.5 km with a standard deviation of 0.8 km. A midfielder who runs 12 km has

$$z = \frac{12 - 10.5}{0.8} \approx 1.9$$

The same idea, all over the pitch

• Sprint speed and distance covered.
• Passes completed per 90 minutes.
• A team's goals over a whole season.

That last one is worth a closer look. A single match's goals follow the Poisson distribution, which isn't a bell curve at all. But add up 38 matches and the total is very close to Normal. A team averaging 2 goals a game expects 76 in a season, with a standard deviation of about 8.7. So 95% of the time they'll finish with somewhere between about 59 and 93, from nothing more than luck. Adding up lots of small random things tends to produce a bell curve; it's why the Normal turns up everywhere.

Show the mathsThe bell curve's formula, and the season-goals check. Optional.

The Normal distribution with mean \(\mu\) and standard deviation \(\sigma\) has density

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

Converting to a z-score turns any Normal into the standard Normal, with mean 0 and standard deviation 1, which is why one table of percentages works for every case. The 68%, 95% and 99.7% figures are the standard Normal's areas within 1, 2 and 3 of zero.

For the season total, a Poisson with mean 76 has variance 76 too, so \(\sigma = \sqrt{76} \approx 8.7\), and the Normal range is \(76 \pm 1.96 \times 8.7\), about 59 to 93. The exact Poisson answer is 59 to 94. The chance of 90 or more is 6.4% exactly and 6.1% by the Normal: close enough for most purposes. That the sum of many independent random amounts ends up near-Normal is the central limit theorem.

Why it matters

Physical and performance data is everywhere in modern football, and most of it arrives as rankings. The Normal distribution turns a ranking into a measure: how far from typical, in a unit you can compare across players, positions and skills. It's the first question to ask of any standout number: how many standard deviations is that?

Limitations

• Not everything is a bell curve. Goals in a match, or a player's goals in a season, are counts that bunch near zero and have a long tail. The Normal only takes over when you add up many of them.
• The comparison group matters. 34 km/h against a whole league is different from 34 km/h against other wingers, who tend to be quicker.
• Tails can be heavier than Normal. Genuinely extreme players and results can turn up more often than the bell curve predicts.

Try it yourself

Find a stat for your team's squad, such as goals per 90 or distance covered. Work out the mean and standard deviation, then each player's z-score. Who is genuinely unusual, and who just tops a close list?

Further reading

• Seeing Theory: probability distributions, Brown University. Interactive: change μ and σ and watch the bell curve move and stretch.
• Normal distribution: properties, proofs, exercises, StatLect. The full derivations.
• Normal distribution, Wikipedia. The formal definition, and the 68–95–99.7 rule.