# Asymmetries in Football: The Pass—Goal Paradox

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Datasets

#### 2.2. Statistical Analysis

## 3. Results

#### 3.1. More Passes, More Goals

#### 3.2. Asymmetries between the Parts of The Match

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Correlation between the number of completed passes and scored goals. Each point corresponds to a team and the red solid line is the linear regression of the points, which has a slope $m=0.026$, an intercept of $b=0.087$ and a correlation coefficient of $r=0.672$.

**Figure 2.**Number of passes per part of the match. For each team i, in blue, number of passes ${n}_{1}\left(i\right)$ completed during the first half of the match. In red, the number of passes ${n}_{2}\left(i\right)$ completed in the second half. Teams are ordered, from left to right, according to the ranking at the end of the season.

**Figure 3.**Number of goals per part of the match. For each team i, in blue, the number of goals ${m}_{1}\left(i\right)$ scored during the first half of the match. In red, the number of goals ${m}_{2}\left(i\right)$ scored in the second half. Teams are ordered, from left to right, according to the ranking at the end of the season.

**Figure 4.**The cost of a goal, in number of passes. For each team i, in blue, the number of passes per scored goal (${n}_{1}/{m}_{1}$) during the first half of the match. In red, the same ratio in the second half (${n}_{2}/{m}_{2}$). Teams are ordered, from left to right, according to the ranking at the end of the season.

**Table 1.**Average number of passes and goals. Teams are divided into three categories: (1) Teams at the top 4 (T4), which qualified for the European Champions League, (2) the 13 teams at the middle of the ranking (MR) and (3) the three teams that were relegated (RE) to a lower division. Numbers correspond to the mean number of passes and goals per match and their corresponding standard deviation.

Group | Passes | Goals |
---|---|---|

Top 4 (T4) | $573\pm 112$ | $1.71\pm 0.45$ |

Middle Raking (MR) | $450\pm 68$ | $1.21\pm 0.22$ |

Relegated Teams (RE) | $422\pm 31$ | $1.07\pm 0.09$ |

**Table 2.**Statistical differences in passes between groups. Teams are divided into three categories: (1) Teams at the top 4 (T4), which qualified for the European Champions League, (2) the 13 teams at the middle of the ranking (MR) and (3) the three teams that were relegated to a lower division (RE). Each row considers a pair of groups (T4-RE, T4-MR and MR-RE), for which we show the average (± standard deviation) difference in passes across all sampling iteration (1000 in total; see main text for details), as well as the average p-value. The third column shows the percentage of cases in which the statistical comparison between groups rejected the null hypothesis of equal means.

Groups | ${\mathit{\mu}}_{\mathit{diff}}(\pm {\mathit{\sigma}}_{\mathit{diff}})$ | ${\mathit{\mu}}_{\mathit{p}}$-$\mathit{val}$ | ${\%}_{\mathit{sig}}$ |
---|---|---|---|

Top 4 (T4)-Relegated Teams (RE) | 150.46 $(\pm 13.18)$ | 9.56$\times {10}^{-10}$ | $100\%$ |

Top 4 (T4)-Middle Ranking (MR) | 121.66 $(\pm 16.28)$ | $9.39\times {10}^{-09}$ | $100\%$ |

Middle Ranking (MR) - Relegated Teams (RE) | 28.79 $(\pm 9.52)$ | $0.19$ | $0\%$ |

**Table 3.**Statistical differences in goals between groups. Teams are divided into three categories: (1) Teams at the top 4 (T4), which qualified for the European Champions League, (2) the 13 teams at the middle of the ranking (MR) and (3) the three teams that were relegated to a lower division (R). Each row considers a pair of groups (T4-RE, T4-MR and MR-RE), for which we show the average (±standard deviation) difference in passes across all sampling iteration (1000 in total; see main text for details), as well as the average p-value. The third column shows the percentage of cases in which the statistical comparison between groups rejected the null hypothesis of equal means.

Groups | ${\mathit{\mu}}_{\mathit{diff}}(\pm {\mathit{\sigma}}_{\mathit{diff}})$ | ${\mathit{\mu}}_{\mathit{p}-\mathit{val}}$ | ${\%}_{\mathit{sig}}$ |
---|---|---|---|

Top 4 (T4)-Relegated Teams (RE) | 45.29 $(\pm 9.56)$ | $0.01$ | $79.4\%$ |

Top 4 (T4)-Middle Ranking (MR) | 35.29 $(\pm 12.81)$ | $0.09$ | $33.5\%$ |

Middle Ranking (MR)-Relegated Teams (RE) | 10 $(\pm 8.7)$ | $0.64$ | $0\%$ |

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## Share and Cite

**MDPI and ACS Style**

Antequera, D.R.; Garrido, D.; Echegoyen, I.; López del Campo, R.; Resta Serra, R.; Buldú, J.M.
Asymmetries in Football: The Pass—Goal Paradox. *Symmetry* **2020**, *12*, 1052.
https://doi.org/10.3390/sym12061052

**AMA Style**

Antequera DR, Garrido D, Echegoyen I, López del Campo R, Resta Serra R, Buldú JM.
Asymmetries in Football: The Pass—Goal Paradox. *Symmetry*. 2020; 12(6):1052.
https://doi.org/10.3390/sym12061052

**Chicago/Turabian Style**

Antequera, Daniel R., David Garrido, Ignacio Echegoyen, Roberto López del Campo, Ricardo Resta Serra, and Javier M. Buldú.
2020. "Asymmetries in Football: The Pass—Goal Paradox" *Symmetry* 12, no. 6: 1052.
https://doi.org/10.3390/sym12061052