Referee bias in Scottish football

A statistical analysis of red cards, 2016-2024

Abstract

In a recent paper, Professor Michael A. Garrett of University of Manchester conducted an analysis of penalty kicks conceded in the top tier of the Scottish Professional Football League (SPFL) in which he concluded that referees were biased towards Rangers Football Club giving them “privileged treatment”. When considering how a referee could realistically be biased towards a team, we assume the majority of football fans would recognize the easiest way to cheat would be:  1) Penalty kicks for and against. 2) Red cards for and against. Whilst a penalty kick offers the chance of a goal, there is no certainty the opposing team will take that opportunity and any advantage ends immediately.  A red card on the other hand has an absolute outcome in that a particular team will play the rest of the match with 1 less player in the team.  This study presents a statistical analysis of ALL red cards in the top tier of the Scottish Professional Football League (SPFL) from seasons 2016/2017 - 2023/2024. In particular, claims online that the number of red cards given to Glasgow Celtic players is statistically anomalous are considered. A test analysis of the red cards given to Glasgow Celtic players versus other SPFL premier league clubs, reveals a statistically significant difference, with Celtic having fewer red cards. This result is even more pronounced for Celtic if the analysis is restricted to red cards at home. By comparison, the number of red cards given to Glasgow Rangers do not diverge significantly from the league's average. The study concludes that the number of red cards given to Celtic over this period is formally a statistical outlier, lending significant weight to concerns over the impartiality of match officiating in Scottish football.