Introduction

The penalty shootout has been in the game of football since the sport has existed. It is used to

decide a winner when the scores are tied at the end of regulation time or it is considered a free

kick when a handball or a foul is made inside the box. Though it is often criticized, nobody can

come up with a better alternative. The penalty kicks in football inspired many other versions of it

in many other sports, like ice hockey or handball. By studying past penalty shootouts, a person

can figure out where is the most successful spot to place a kick and where the goalkeeper will

most likely dive.

Study Aim

For my Internal Assessment, I will be studying penalty kicks in football. I will specifically be

looking at success percentage in nine different areas of the goal, middle, top middle, bottom

middle, left middle, bottom left, top left, middle right, top right, and bottom right. The area of

math that this topic is placed into is probability and statistics. I have been playing football since

the age of 5 years old being mostly goal keeper. I chose this topic because I believe that if I learn

where the highest probability of scoring a penalty kick is then I will be able to apply it to myself

and improve as a player and even to help other players improve their skills through a

mathematical exploration.
Reasearch Questions

1.

2. Which area of the post ensures a higher rate of scored goals in penalties?

3. Which cell should the goal keeper dive into in order to maximize his chances of catching

the ball?

4. How do my results compare with the Telegraph study (cell-wise)?

Assumptions

Wind speed not taken into account

Air resistance not taken into account

No tricks

No crowd pressure, coach pressure

Reasearch Methodology

I have collected my data about penalty kicks through a realistic football game, FIFA 17. I have

selected 2 players, one left-footed and one right-footed – both midfielders with an average rating

(i.e. 82-84 %, best player having a 94 % rating). The players are called Player X and Player Y.

Using a simulation module of FIFA 17, I have kicked 100 penalties for each player, using the

same goal keeper for both players

First Model

In football, the goal post has the following dimensions:

approximately in length.

I have noted the rate of successful goals in each of the 3 areas.

Goal Keeper Dives

NS C OS
Kicker shoots

NS 50 70 70

C 62 0 62

OS 65 65 47

The NS is the natural side of the kicker (for example, a right footed player’s natural side is the

right side of the goal keeper)

The C is the centre of the goal post

The OS is the opposite side of the kicker (for example, a right footed player’s opposite side is the

left side of the goal keeper).

My results suggest that:

1. The highest probability of success is when kicker shoots with his NS and goal keeper

stays in centre or dives to the OS. This makes sense, as the kicker is shooting on his

preferred side and the goalkeeper stayes on center or dives in opposite side, hence he

cannot catch the ball easily.

2. The lowest probability of success (apart the Center – Center case)is when kicker uses his

OS and the goal keeper dives on same side. This is due to the fact that kicker is not very

performant on his OS and the goal keeper dives the right direction.
 3. In between, we have the situations when kicker shoots in the center. Here the chances of

scoring are 0 when goal keeper stays in the center, and are quite small when goal keeper

goes to his right or left.

4. When kicker uses his opposite side or shoots in the center, the chances are 65 % of

scoring, greater than in case 3.

Refined Model

I have realized that my first model does not take into account the different heights the kicker can

shoot at. So I came up with a refined model, splitting the goal post into 9 cells as follows:

Left Up Center Up Right Up

Left Center Center Right Center

Left Down Center Down Right Down

Post split into 9 equal area cells.

Drawing with the post H-2.44 m, W=7.32 m

The Telegraph study showed:

footed player. For a more unbiased result, I decided to use two average players, Granit Xhaka

from Arsenal, Rated 84 OVR, as my left-footed shooter and Gonzalo Castro from Borussia

Dortmund, Rated 82 OVR, as my right-footed penalty taker.

S0% S1% S1%

M 16 % M 22 % M 15 %

G 8% G 5% G9%

S4% S5% S 11 %

M6% M0% M 12 %

G 20 % G 13 % G 15 %

S 16 % S 20 % S 33 %

M 24 % M0% M5%

This results suggest that:

S0% S1% S1%

M 16 % M 22 % M 15 %

G 8% G 5% G9%

S4% S5% S 11 %

M6% M0% M 12 %

G 20 % G 13 % G 15 %

S 16 % S 20 % S 33 %

M 24 % M0% M5%

Comparison

t-tests were performed to see how far our result correlate with the Telegraph study.

About t-tests

Conclusion

From the data I have collected, the best spot to aim at is straight down the middle. This may not

be true every time though. Also from the data I have collected, the worst spot to aim when taking

a penalty kick is at the bottom middle of the goal. This is so because a keeper will still have his

legs in that spot for a second when he starts to dive one direction, and the ball will probably take

less than a second to get to the goal line. The bottom left and right of the goal is where most
players will aim the ball when taking a penalty kick, 94 times in total. This also means that the

keeper will dive to that position most often. The middle left and middle right of

the goal also contribute to quite a few penalty shots, where the keepers will extend an arm and

save the shot when they dive to a corner.

Limitations:

1.Borderline Decisions: Some of the penalty shots were arguable on whether they were in a

certain position of the goal. Such as, the player shoots the ball but someone could argue that is

was in the top left while another person argues that the shot was in the middle left. So I had to

make a decision where the goal would be placed. Using one or 2 coins.

2. Valid Penalty: Another decision I had to make was whether the shot would be considered

saved if the ball hit the frame of the goal. I have considered it as missed

3.How many: How many penalty kicks should I watch? I gave myself a two hour window to

watch as many penalty shootouts as I could, and I ended up recording 190 penalty kicks. This

might not be enough to have accurate recordings. Sample size problem

http://www.telegraph.co.uk/sport/football/competitions/euro-2012/9361604/The-science-of-

penalty-shoot-outs-where-and-how-spot-kicks-are-scored-at-major-tournaments.html

https://www.scribd.com/document/222781514/ib-math-sl-internal-assessment

http://www.dailymail.co.uk/sciencetech/article-2661383/The-science-penalty-kicks-Size-goal-

reaction-times-game-theory-odds-overwhelmingly-favour-shooter.html