# A midfielder's match in five numbers. Vectors

Source: https://www.footballdatascience.co.uk/learn/player-as-a-vector
Published: 2026-09-27

> A player's performance can be written as a list of numbers in a fixed order. That list is a vector, and it's the first step to comparing players, finding replacements and feeding football into machine learning.

## The football question

A central midfielder has just played a full match. **How do you describe his performance in a way a computer can compare with every other midfielder?**

Match reports use words: "tidy", "busy", "got about the pitch". Words are hard to compare. Numbers aren't.

## The concept

A **vector** is an ordered list of numbers. Each position in the list always means the same thing, so two vectors can be compared position by position.

Here is our midfielder's match:

| Passes | Tackles | Shots | Chances created | Distance (km) |
|---|---|---|---|---|
| 68 | 7 | 3 | 5 | 11.2 |

Written as a vector:

$$\mathbf{v} = (68,\ 7,\ 3,\ 5,\ 11.2)$$

<div class="plain" markdown="1">
In plain football

- **Bold v** is the name of the vector: this midfielder, this match.
- **Five numbers** means five *dimensions*. The first is always passes, the second always tackles, and so on.
- **Order matters.** (7, 68, …) would be a completely different player: seven passes and 68 tackles.
</div>

That's all a vector is. The power comes from what you can do with it.

## Things you can do with a vector

### Scale it: per 90 minutes

Suppose he was substituted after 75 minutes. To compare him fairly with players who played the full 90, scale everything up by 90 ÷ 75 = 1.2:

$$1.2 \times \mathbf{v} = (81.6,\ 8.4,\ 3.6,\ 6,\ 13.4)$$

<div class="plain" markdown="1">
In plain football

- Multiplying a vector by a single number multiplies **every** entry by it. This is called *scalar multiplication*.
- It's exactly how "per 90" stats are made: at this rate, over a full match, he'd have made about 82 passes and 8 tackles.
- Per-90 figures can flatter players who only play short spells, often late on when the game is stretched, so treat them with care.
</div>

### Add them: a run of matches

His next match looks like this: 54 passes, 4 tackles, 1 shot, 3 chances, 10.6 km. Add the two vectors entry by entry to get his totals:

$$\begin{aligned} &(68,\ 7,\ 3,\ 5,\ 11.2) \\ +\; &(54,\ 4,\ 1,\ 3,\ 10.6) \\ =\; &(122,\ 11,\ 4,\ 8,\ 21.8) \end{aligned}$$

<div class="plain" markdown="1">
In plain football

- **Vector addition** adds passes to passes, tackles to tackles, and so on. It never mixes them.
- Divide the total by 2 and you have his average match: (61, 5.5, 2, 4, 10.9).
- A season is the same idea with 38 vectors instead of two.
</div>

### Subtract them: the difference between two players

Another midfielder has (52, 3, 1, 2, 10.1). Subtract his vector from ours:

$$\begin{aligned} &(68,\ 7,\ 3,\ 5,\ 11.2) \\ -\; &(52,\ 3,\ 1,\ 2,\ 10.1) \\ =\; &(16,\ 4,\ 2,\ 3,\ 1.1) \end{aligned}$$

<div class="plain" markdown="1">
In plain football

- Each entry is how much more of that thing our midfielder did.
- **16 more passes** looks like the biggest gap. It isn't. That's 31% more passes, but 4 more tackles is more than twice as many.
- The numbers are on different scales: a midfielder makes dozens of passes and a handful of tackles. Comparing raw gaps lets the biggest numbers shout loudest.
</div>

<figure class="rank-chart">
<div role="img" aria-label="Two bar charts of the gap between the two midfielders. Raw: 16 passes, 4 tackles, 2 shots, 3 chances, 1.1 km. As a percentage: passes +31%, tackles +133%, shots +200%, chances +150%, distance +11%.">

</div>
<figcaption>The same five gaps, two ways. Raw, passes dwarf everything because players make dozens of them. As a percentage of the other player's figure, the tackles, shots and chances gaps are far bigger.</figcaption>
</figure>

That last point matters for everything that follows in this series. Before comparing players, analysts usually put every entry on the same scale, often by turning each into a z-score: how many standard deviations above or below average it is, as in [the Normal distribution](/learn/normal-distribution). Then a big pass count and a big tackle count count the same.

<details markdown="1">
<summary><span>Show the maths<small>The general definition, and why the operations work entry by entry. Optional.</small></span></summary>

A vector with *n* entries is written

$$\mathbf{v} = (v_1,\ v_2,\ \ldots,\ v_n)$$

and lives in *n*-dimensional space, \(\mathbb{R}^n\). Our midfielder is a point in \(\mathbb{R}^5\). Vectors are also often written standing up, as a column:

$$\mathbf{v} = \begin{pmatrix} 68 \\ 7 \\ 3 \\ 5 \\ 11.2 \end{pmatrix}$$

Addition and scalar multiplication work entry by entry:

$$\mathbf{a} + \mathbf{b} = (a_1 + b_1,\ \ldots,\ a_n + b_n)$$

$$c\,\mathbf{v} = (c\,v_1,\ \ldots,\ c\,v_n)$$

The average of *m* vectors is their sum times \(1/m\), which is why a season average is just addition followed by scaling.

</details>

## Why it matters

Once a performance is a vector, a computer can do things a scout can't do by eye across thousands of players:

- **Compare** two players number by number.
- **Measure similarity**: which midfielders play most like ours?
- **Find replacements**: who is closest to the player we're about to sell?
- **Group players by style**: ball-winners, creators, box-to-box runners.
- **Feed machine learning models**, which expect every example as a vector of numbers.

All of those need a way to measure how far apart two vectors are, which is where this series goes next.

## Limitations

- **A vector only knows what you put in it.** Five numbers say nothing about positioning, decision-making or leadership. Choose the entries carefully.
- **Counts depend on context.** A midfielder in a side that has the ball all game will make more passes, whatever his quality.
- **Units and scales differ.** Kilometres and pass counts can't be compared raw, as the subtraction showed.

## Try it yourself

Pick two midfielders from your team and write down the same five numbers for each from their last match: passes, tackles, shots, chances created and distance. Scale each to 90 minutes, then subtract one vector from the other. Which gap looks biggest, and which is biggest once you think in percentages?

## Further reading

- [Vectors, what even are they?](https://www.3blue1brown.com/lessons/vectors), 3Blue1Brown. The best visual introduction there is; the first chapter of *Essence of linear algebra*.
- [Vectors and spaces](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces), Khan Academy. Short lessons with practice exercises.
- [Vectors and matrices](https://www.statlect.com/matrix-algebra/vectors-and-matrices), StatLect. The formal definitions, in the notation used in statistics.