Fouls are an integral part of soccer matches and occur frequently in matches (Gümüşdag et al., 2011). Since 1863, when The Football Association in England published the first Laws of the Game, officially defining football rule and establishing punishments for fouls (e.g., handball, violence), referees have served as the primary enforcers of these laws and the decision-makers for fouls committed by players (Curry, 2020). According to the Laws of the Game, fouls can be categorized in two types: those punishable by direct free kicks and those resulting in indirect free kicks (The International Football Association Board (IFAB), 2024). Number of fouls refers to the total fouls committed by both teams in a game (Yuan et al., 2024). Studies indicate that during the 2023–2024 season, the England Premier League (EPL) averaged 21.5 fouls per match while the German Bundesliga reported a slightly higher average of 21.9 fouls (Fournier, 2024). Fouls are often used tactically to disrupt opponent's counterattack or promising attacks (Englund, 2022; Lepschy et al., 2018). However, excessive fouls can lead to frequent match stoppages and even escalate into physical fights between teams (Joseph, 2015; Fields et al., 2009).

Previous studies have identified some factors influencing foul frequency. For example, Nevill et al. (2002) and Unkelbach and Memmert (2010) found that crowd noise significantly influenced referees' decisions, while Lex et al. (2015) demonstrated that players' vocalizations had no significant influence on foul calls. In addition, home advantage refers to the consistent finding that home teams win more often than away teams (Lago-Peñas and Lago-Ballesteros, 2011), and has been linked to foul distribution. For instance, Goumas (2012) reported that home teams received fewer fouls than away teams. The COVID-19 pandemic provided a unique opportunity to examine the influence of audiences on home advantage. Multiple studies reported that home advantage declined, with the number of fouls for home teams significantly increasing during matches played without spectators (Endrich and Gesche, 2020; Leitner and Richlan, 2021; Webb, 2021). In addition, Han et al. (2020) found that the number of fouls significantly declined after the application of the Video Assistant Referee system (VAR) in the Chinese Super League (CSL). Therefore, these findings highlight the need to further investigate the effects of air temperature and explore additional factors, including team strength disparities, game stages, and referees' running performance, on the number of fouls.

A soccer match is officiated by a group of referees, consisting of one Main Referee (hereafter referred to as “the referee”), two Assistant Referees (AR), and a fourth official [FO; (The International Football Association Board (IFAB), 2024)]. Moreover, Video Assistant Referees (VAR) and an Assistant VAR (AVAR) are used as additional support referees in some professional soccer matches such as the EPL, La Liga, the German Bundesliga, Serie A, the FIFA World Cup, and the Chinese Super League (CSL; Han et al., 2020). During a match, other referees (e.g., ARs, FO, VARs) have the right to provide information and advice to the referee on some foul situations, especially in relation to major or controversial fouls [e.g., penalties, yellow and red cards; (The International Football Association Board (IFAB), 2024)]. However, all fouls must be decided by the referee after running and observing the match (Kranjec et al., 2010). Consequently, referees, especially the referee, need to have an excellent level of physical fitness and their fitness test scores are important criteria for referees' promotion and qualification (Chiu, 2022). In this study, the running performance of the referee refers to the referees' total running distance throughout the game and the distance between the referee and the location of the foul when adjudicating it (Riiser et al., 2019). Referees are encouraged to run actively and minimize the distance from the referee to the foul incident location when making decisions (Mallo et al., 2010). Johansen and Erikstad (2021) found that decision accuracy is highest when referees' decision distance is within 10 m. With high decision accuracy, players may not feel angry about the judgment and are less likely to commit fouls as revenge. In addition, the active movement of the referee can create a sense among players that they are being closely monitored, potentially reducing the occurrence of unnecessary infringements. Referees are also expected to verbally warn players before a potential foul (Mascarenhas et al., 2006). Therefore, the referees' running performance (i.e., active ball running distance, decision-making distance) during a match may influence the occurrence of fouls. However, the relationship between referees' running performance (i.e., running distance and decision-making distance) and the number of fouls remains unclear.

Team quality is represented by the team's ranking at the end of the season, which has been commonly used in soccer tactical and technical analysis (Kong et al., 2022; Villaseca-Vicuña et al., 2024). The difference in team quality may be a significant factor affecting the number of fouls in soccer matches. Lago-Peñas (2012) demonstrated that soccer players' running performance can be influenced by the quality of the opposition. The contest for ball possession may become intense, resulting in more physical contact and a higher number of fouls when the two evenly matched teams compete. Previous studies have reported that the weaker team might commit more tactical fouls to restrict or hinder their opponents' attack (Liu et al., 2016). However, there is an extremely limited amount of research examining the impact of the difference in team quality on the number of fouls in soccer matches to date.

The number of fouls may also vary under different air temperatures during the game. Guy et al. (2015) found that running performance in elite middle- and long-distance athletes was negatively impacted at air temperatures above 25°C. The extent of performance decline correlates with both the duration of the event and the magnitude of temperature deviation from the optimal range of 10–15°C for endurance performance (Ely et al., 2007). In soccer matches, high-speed running, sprinting, tackling, accelerating, and decelerating occur intermittently, requiring a higher level of physiological and neuromuscular capacity (Harper et al., 2019; Stewart et al., 2019). Motion analysis data from the 2014 FIFA World Cup revealed a significant decline in high-intensity activities (e.g., sprinting) and high-intensity running distance at 34°C compared to 19°C (Nassis et al., 2015). Laboratory experiments have shown that increasing core and muscle temperatures result in approximately a 10% reduction in repeated-sprint performance (Drust et al., 2005). This impairment is consistent across both dry and humid heat conditions (Hayes et al., 2014). Kang et al. (2024) compared football players playing at temperatures under 21°C and over 29°C and reported that higher heart rate responses were found for players when playing at temperatures over 29°C, while more successful passes were observed for players when playing at temperatures under 21°C. Consequently, in high air temperature environments, the compromised physical ability may lead to fewer physical contacts and fouls. For example, Yuan et al. (2024) initially explored the effects of weather conditions on the number of fouls and found that air temperature and precipitation could influence the number of fouls. The study showed that there was a 7.6% decrease in players' running distance and a 23.1% decline in the number of fouls committed by CSL players in games played at temperatures over 30°C compared to games played at temperatures under 15°C. Although Yuan et al. (2024) study pioneered the research on the effects of temperature on the number of fouls, a significant limitation remains. All the data were collected from the CSL from 2015 to 2017, when there was no VAR. Therefore, it is crucial to explore the number of fouls across different temperatures under prevailing conditions (with VAR).

A regular season usually lasts more than 6 months, which could be equally divided into the early stage, middle stage, and the last stage (Zhou et al., 2019). The average number of fouls may vary across different stages. For example, in the early stage of the season, players are expected to be equipped with ample physical and psychological preparation after the off-season (Alexander, 2020). Players vigorously participate in body confrontation and challenges, which may lead to an increase in fouls. As the league progresses, players gradually adapt to the match rhythm, and their physical and psychological condition may decline due to the cumulative effects of earlier matches, possibly resulting in fewer fouls compared to the season's outset. While previous studies have demonstrated differences in football players' running and technical performance in different air temperatures and stages (Chmura et al., 2021; Li et al., 2023; Özgünen et al., 2010), no study has examined the variations in the number of fouls under these two conditions.

Based on the previous review, we found that the referee's running performance, team quality difference, temperature, and league stages may influence the number of fouls. Therefore, the purpose of this study was to (1) examine the impact of referees' running performance during matches (running distance of the referee, decision distance of the referee) and the difference in teams' quality on the number of fouls; (2) explore whether there are significant differences in the number of fouls under various air temperature conditions and at different stages of the season. The findings of this research could academically advance our understanding of the factors that impact the number of fouls in soccer games. Particularly, we explore how referees' running performance, team quality difference, game stage, and air temperature influence the number of fouls in soccer matches. The findings of this study provide valuable practical implications for referees, thereby helping them predict the frequency of fouls, develop effective running strategies to reduce the number of fouls, and finally increase the smoothness of the game.

This study examined how the referees' running performance, difference in team quality, temperature, and league stages affect the number of fouls in soccer matches. Our findings contribute academically to extending the understanding of foul occurrences in soccer. From a practical perspective, the results provide valuable information for referees to evaluate games and formulate strategies to reduce the number of fouls and improve game quality.

The current study uncovered a significant negative correlation between referees' running distance and the number of fouls (r = −0.279, p < 0.01). This finding aligns with our hypothesis and supports the strict physical requirements set by the referee administration department for referees (Chiu, 2022). The regression model further provided evidence for this relationship: Number of fouls = 45.801 – 0.002 ^* Referee running distance (SEE = 6.351). The results show that if referees run an additional 1 kilometer during the game, the number of fouls decreases by 2. Although the effect size is small, it highlights the potential impact of referee movement on the course of the game. This finding can be explained in several ways. Firstly, active running during the game provides referees with a better viewing angle, improving the accuracy of calls and convincing players (Mallo et al., 2010). In addition, active running creates more opportunities for referees to communicate with players (Simmons, 2006), thereby alleviating player dissatisfaction with the referee in a timely manner and defusing potential conflicts between players, which in turn helps to reduce subsequent retaliatory fouls arising from anger.

Surprisingly, no significant correlation was found between the number of fouls and either the decision distance of the referee or the strength difference between teams, which is not aligned with our initial hypothesis. The lack of correlation between decision distance and fouls could stem from the non-linear relationship between decision distance and accuracy. While Mallo et al. (2012) and Hossner et al. (2019) suggested optimal decision distances of 10–15 m, where referees make fewer errors, De Oliveira et al. (2011) found no significant relationship between the decision distance of the referee and foul accuracy when using continuous numeric variables to represent the running distance of referees. Similarly, Riiser et al. (2017) also reported that the decision accuracy is not related to referees' position in relation to the infringements. Therefore, we may assume that despite referees being in close proximity, players may still commit fouls as a form of retaliation, if the referee's previous decision was perceived as incorrect (Mark et al., 1983).

The absence of a significant relationship between the strength difference between teams could be explained by the fact that season-end rankings may not accurately represent the importance of a game at a specific time. For example, a team facing imminent relegation may play aggressively against other teams (e.g., with frequent body contacts and fouls) in the first stage, as they are eager to earn points. In contrast, the same team might play more relaxedly and with less intensity (e.g., fewer body contacts and fouls) toward other teams when the team is doomed to be relegated in the last stage, even when the teams' strength difference remains close at the end of the season.

Our study also found a significant difference in the number of fouls across varying temperature conditions (F = 6.669, p < 0.001). In particular, the number of fouls was more prevalent in cooler temperatures compared to hotter and warmer temperatures. Previous research has shown that temperature plays a role in influencing athletes' physical performance (Guy et al., 2015). In hot environments, athletes experience heightened metabolic rates, which can reduce their explosiveness and endurance, potentially impacting athleticism and aggressiveness (Harper et al., 2019; Stewart et al., 2019). This may reflect a mechanism by which the number of fouls is indirectly influenced by air temperature through athletes' performance conditions. When the temperature is cool, players' mental and athletic performance is optimal (Kang et al., 2024), leading to increased body contacts and consequently a higher number of fouls. Conversely, when the air temperature is hot or very hot, athletes' mental performance decreases and their physiological functions decline (Kang et al., 2024), resulting in a reduction in the number of fouls.

Furthermore, our findings supported the hypothesis that the number of fouls in Stage 1 (1–10 rounds) was significantly higher than in Stage 2 (11–20 rounds; p = 0.002). This result can be attributed to fluctuations in physical and psychological conditions. Carling et al. (2015) noted that football players exhibit decreased physical and technical performance toward the end of the season. In contrast, players are typically well-prepared in terms of fitness and mindset at the season's commencement, potentially leading to increased body contact and confrontations, resulting in more fouls. After experiencing 10 rounds, players' physical fitness drops, and motivation wanes, resulting in fewer fouls.

Based on our findings, we recommend that referee management departments continue to stress the importance of referees' fitness and advocate the role of referees' running performance in reducing the occurrence of fouls. Referees' running distance can be acknowledged as one of the criteria for promotion and rewards. In addition, referees are encouraged to be carefully prepared both mentally and physically when officiating matches under 20°C and at the beginning stage of the season.

Although this study significantly contributes to foul studies in football literature, several limitations remain. First, the data used was exclusive from the CSL, therefore limiting the generalizability of the findings. Moreover, different referees may have varying standards for foul commitment, which may undermine the validity of the data. Therefore, future studies may include soccer games outside of China (e.g., European soccer leagues and World Cup). Third, the air temperature was categorized according to James et al. (2022) criteria, which was conducted in Malaysia where the athletic environment is different from China (e.g., humidity and altitude), consequently, future studies may reference environmental criteria based on local studies. Fourth, we evidenced running distance negatively associated with number of fouls, however the effect size is small. In addition, decision distance and game stage were found not to relate to the number of fouls. Consequently, more factors, especially contextual factors, are encouraged to be included in future studies. Building on Goumas's (2012) finding that crowd density could result in referees' bias, future studies may incorporate stadium attendance and running track to explore environmental effects on the number of fouls in soccer games.

This study revealed that referees' running distance in games negatively relates to the number of fouls. Additionally, more fouls occurred when games were played in cool environment situation and in the early stage of the season. No significant relationship was found between the number of fouls and either the decision distance of the referee or the strength difference between teams. The finding not only pioneers research on the number of fouls from the perspective of referees' running performance, weather conditions, and stage, but also extends the understanding of the patterns of foul occurrences. The findings of the study also offer practical implications for referees to predict the frequency of fouls, thereby enabling better game preparation based on weather conditions and the stage of the game. The negative relationship between running distance and number of fouls provides insights into running strategies to reduce foul occurrences.