Math is of incredible importance in our society. Of course, most will not use **the Pythagorean theorem ** on a daily basis. Nor will they have to solve **quadratic equations** before breakfast. But many will acknowledge that math is all around us, even where we least expect it. And one such place is football.

I regularly enjoy a game of football. And I will be the first one to admit; when watching Ajax, I don’t use mathematics that much. Maybe just to calculate the risk of missing a goal while running to the fridge for a cold one. And even then, my calculations might be somewhat off, and also have relatively little to do with actual math. So, what *does *math have to do with football?

#### ExpG

Sander IJtsma, a data-analyst and the man behind the twitter account **@11tegen11 **, is the creator of the Expected Goals (ExpG) model. Historically, goals and match points were the only statistics of relevance. If you wanted to see how a team was performing, you would examine the league table. When more data became available, statistics such as total shots and shots on target gained interest. The latest trend in this field is the ExpG model. This model gives all attempts a value between 0 and 1, reflecting the odds of that attempt turning into a goal. Factors of influence for this value are, amongst others, shot location and shot type. But next to factors relevant to the situation on the field, the model also makes use of databases to determine the correct odds by regression methods and earlier shots.

The ExpG model can be used to predict the future points of a team in a season, or the future goal ratio. These two graphs show the differences between the different predictors:

When comparing this model to the other statistics previously used, it immediately shows that the ExpG is far superior. It is accurate and ready to be used earlier than its older counterparts (ExpG is useable after around 4 games, whereas points per game and goal ratio take around 10 games to become useful), and although the shots on target ratio and total shots ratio also become useful quite early on, these ratios do not score as well as ExpG overall, and they also lose their predictive capacities earlier than the ExpG.

So, let’s dive into the model a bit deeper; Sander IJtsma uses the penalty to explain his model:

*“Typically, penalties are awarded around 0.76 ExpG, based on historic conversion rate. A penalty is the easiest attempt to classify, since it’s a situation isolated from play, with a standard spot for taking it. The number of penalties taken is way too low to factor in player or keeper performance, so we do best by just estimating 0.76 ExpG.”*

The model defines 10 situations, amongst which the penalty, open play shots, open play headers, direct free kicks and rebounds from a goalkeeper save (for all situations and more information, click here). All situations have a separate regression model, since a free kick obviously has different factors that influence its ExpG value than a rebound from a goalkeeper save.

For each of these ten situations, the model incorporates several factors on which the ExpG value is based. These factors include, amongst others, shot location, shot type, big chance and game state. Shot location is the most important factor in ExpG, and is based on the angle of view of the goal and the distance from the goal. The shot angle is measured through two lines from the shot location to the posts of the goal. The angle between these two lines signifies the angle the player has.

All of this data is objective, without interference from people who might be prejudiced towards a certain outcome. However, the data itself is not enough to aptly judge a situation. This is where the big chance factor comes into play. When a player takes a shot from really far away, the shot location will give the attempt a very low value. However, if that attempt would occur because the goalkeeper is out of position (for example when there is a corner at the last minute and the keeper comes forward as well, a counter by the opposition would mean the goalkeeper is not in his goal yet), then the attempt would have a much higher value. This is accounted for by the big chance factor, which is manually added by professional coders after assessing the situation. The last factor I want to briefly discuss is that of game state. The game state refers to the state of the game at the moment of the attempt (who would have guessed!). For example, when a team is winning, they will defend differently from when they are losing. As a result, scoring is easier or harder, which thus affects the ExpG.

Okay, let’s show an example to clarify:

IJtsma elaborates on this specific example with all the relevant values which together create its ExpG value:

- Situation: Indirect Free Kick
- Shot location: Angle of view 10.4 degrees and distance 34.1.
- Shot type: foot shot
- Big Chance: no
- Start of possession: no high turnover
- Assist: unintentional
- Through ball: no
- One pass after a through ball: no
- Cross: no
- Dribbles: 0
- Dribbles around the keeper: 0
- Vertical speed: 2.05 meters per second
- Number of Touches: 2
- Game State: 0

**ExpG value: 0.013**

Everybody can see that this is a beautiful goal, and that it is something special. But how rare this exactly, is hard to explain. This ExpG value means that it would take around 75 attempts for one of these shots to go in.

It is statistics like these that nowadays are incorporated by many coaches, clubs and analysts. Influential coaches like Pep Guardiola, Thomas Tuchel and Jürgen Klopp are incorporating math and statistics in their tactics. There even are clubs that solely rely on statistics and models like these to manage their teams; Brentford (second tier English football team) and FC Midtjylland (first tier Norwegian football team).

#### SOWISO

Math and statistics are up and coming in the football field, and they are already playing important roles in baseball (the movie “Moneyball” is an excellent example) and basketball. Math is all around us, and it has more awesome features than it gets credit for. So whether you are having trouble with math, simply want to learn more about it or become even better at it, SOWISO and PassYourMath are most definitely the platforms to use to improve!

Thank you for reading, I hope you enjoyed reading this blog as much as I did writing it. If you have any questions or comments, feel free to drop them below!

Thijs

Hi, thanks for sharing this article and the video. It’s fascinating to see how statistics can be used in sports and in team management. I’m thinking that some of my high school math students would be hooked with this type of example of how math can be used in their world.

Hi Susan, good to hear you liked the article! Feel free to use it in class!