By Tony Attwood

Preliminaries: this article uses the same figures as used in articles from a couple of weeks back concerning tackles, fouls and yellows. This means the numbers here are a few weeks out of date – however I have no reason to think that updated numbers would show a different set of relationships. The only reason for not updating them is that the whole process takes quite a bit of time – which I don’t have at the moment.

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This short article concerns a question that has been sent in a large number of times in comments relating to earlier posts about the way referees are handling individual teams.

It shows in column three how many tackles clubs have to make in order to have a foul called by the referee. In column four it shows how many fouls are called before a yellow card is given.

Pos | Team | Tackle/ Fouls | Fouls/ Yellows |
---|---|---|---|

1 | Liverpool | 1.90 | 8.33 |

2 | Manchester City | 1.43 | 5.21 |

3 | Leicester City | 2.08 | 9.48 |

4 | Chelsea | 1.82 | 5.31 |

5 | Tottenham Hotspur | 1.79 | 4.46 |

6 | Sheffield United | 1.61 | 5.36 |

7 | Manchester United | 1.45 | 5.68 |

8 | Wolverhampton W | 1.63 | 6.82 |

9 | Everton | 1.60 | 6.55 |

10 | Arsenal | 1.61 | 4.32 |

11 | Burnley | 1.29 | 6.14 |

12 | Newcastle United | 1.81 | 5.64 |

13 | Southampton | 1.67 | 7.73 |

14 | Crystal Palace | 1.67 | 6.50 |

15 | Brighton and Hove A | 1.76 | 6.92 |

16 | AFC Bournemouth | 1.55 | 4.62 |

17 | Aston Villa | 1.39 | 5.89 |

18 | West Ham United | 1.88 | 5.61 |

19 | Watford | 1.55 | 5.30 |

20 | Norwich City | 1.54 | 4.83 |

We can notice that Leicester City have to undertake 2.08 tackles per foul, compared with Arsenal players who commit 1.61 tackles before a foul is called. But the big difference comes from the number of fouls a club commits to get a yellow card. For Leicester it is 9.48, for Arsenal it is 4.32.

With a set of figures like this mathematicians usually then look at the standard deviation. This is a number used to indicate how measurements for a group are spread out from the average (mean), or expected value. A low standard deviation means that most of the numbers are close to the average. A high standard deviation means that the numbers are more spread out.

This then is a way of answering the question that has been put an increasing number of times. How are the numbers in the tackles to fouls and fouls to yellows related to each other?

The Tackles per Foul Standard Deviation is **0.19** – a low figure showing that there is a close relationship across the board between the way referees award fouls following tackles. By and large clubs are being treated in the same way as each other – with the only key outlier being Leicester City. They are being treated very differently from every other club as the numbers show.

However the fouls per yellow Standard Deviation **1.29 – **over six times the SD for tackles per foul. This is where clubs are being treated very differently.

This, however, does not mean that there is no relationship between the number of fouls and the number of yellow cards. All the numbers lie in a band between 4.62 and 9.48 with an average of 6.04.

But this does show that there is none of the consistency in handing out yellow cards which is to be found in calling fouls. However it does not mean there is no relationship between fouls and yellows. There is a relationship, but it is far more spread out – this is where referees are using their leeway and where a biased referee would have a great opportunity to influence the game as he or she wishes.

The reason for this comes from the way the Laws of the Game are written.

A list of specific offences that can be fouls are detailed in Law 12 of the Laws of the Game and these offences are clear

- Kicks or attempts to kick an opponent
- Trips or attempts to trip an opponent
- Jumps at an opponent
- Charges an opponent
- Strikes or attempts to strike an opponent
- Pushes an opponent
- Tackles an opponent
- Holds an opponent
- Impeding the progress of an opponent with contact
- Spits at an opponent (considered Violent Conduct as the spit is considered an extension of the body)
- Handles the ball deliberately (except for the goalkeeper within their own penalty area).
^{}

These are all “did he or didn’t he?” type offences. Did he trip an opponent? Yes = foul. No = no foul.

But the yellow card offences are different, and one might argue that they are much vaguer, and this is where referees can manipulate the situation should they have a mind to. Yellows are given for…

- Unsporting behaviour (foul play, simulation, and denying an obvious goal-scoring opportunity by an offence which was an attempt to play the ball and the referee awards a penalty kick are included)
- Dissent by word or action
- Persistent infringement of the Laws of the Game
- Delaying the restart of play
- Failure to respect the required distance when play is restarted with a corner kick, throw-in or free kick
- Entering or re-entering the field of play without the referee’s permission
- Deliberately leaving the field of play without the referee’s permission

You only have to look at the first item in this list to see how vague it is. “Unsporting behaviour” = yellow. But not always!

There are also two offences which apply in matches using the Video Assistant Referee system:

- Entering the referee review area
- Excessively using the ‘review’ (TV screen) signal
^{}

So that’s what the numbers show. The number of fouls is related to the number of tackles very closely, as we would expect. But when it comes to handing out the yellow cards, referees dish these out to teams in very different numbers. There is still a relationship between the number of fouls and the number of yellows but it is not nearly as close as with tackles to fouls.

Clearly, if we want to see if referees are biased or not, we need to be looking at the way yellow cards are handled because although there is a relationship to the number of fouls, it is wider than the tackles to fouls ratio.

I hope that clarifies the situation for the various readers who have written in on this topic.

Thanks for the hard work and research .

Whoscored.com has a table that shows “situational statistics”. For “card situations” it has four definitions, foul, professional, dive and other. Now as no team appears to have received a card for a dive I assume they don’t analyse this for some reason so let’s stick with the other three categories. Now this shows up something very interesting. (I count a red as 2 cards for my calculations)

When you look at cards given for fouls the average is a card every 7.9 fouls. The lowest (most harshly treated) is with a card every 5.6 fouls (Bournemouth). The most leniently treated is (surprise, surprise) Leicester with a card every 11.6 fouls. Arsenal are actually not treated too harshly being only slightly worse than average at a card every 7.1 fouls.

Now let’s look at cards for “unprofessional” (whatever that means). On average each team has had almost exactly 2.5 of those this season (or one every 13.7 games). The most leniently treated is Wolves with no cards but just behind them on just one card is (surprise, surprise) Leicester (along with five other teams it has to be said). The most harshly treated team with 7 cards (one every 3.9 games). Arsenal are also harshly treated being the third most carded team with 4 (one every 6.8 games).

When we look at “other” (not defined by whoscored as far as I can see) it becomes more interesting. I can only assume this is for time wasting, dissent, delaying the start of play etc. In fact all those vague offences that Tony listed above which can be given on a whim since, whilst being actually legitimate, it is these that are the least clear cut reasons for a card and hence somewhat arbitrary and, to some extent, a “choice” of the referee. So…………

On average teams have received 9.9 cards (one every 3.2 games). The least harshly treated is (SURPRISE, SURPRISE, SURPRISE!!!) Leicester…..could this just be one massive coincidence I ask myself?! They received just four cards for “other” (one every 6.8 games). The most harshly treated (just as surprising….NOT) is Arsenal with 20 cards for “other” (one every 1.4 games) and seven more cards than the second most harshly treated team. To give that some context, we get as many cards for “other” as Leicester, Liverpool, West Ham and Southampton put together.

So when we take cards given for “offences” other than a foul Arsenal gets a card every 1.1 games whereas, Leicester get one every 5.4 games (the average is 2.5). This starts to put things in perspective.

The other thing worth noting is that the six most harshly treated teams for receiving cards for “non-foul offences” are all from the south of England. Indeed 7 of the 8 most harshly treated in terms of fouls per card are also from the south. Make of that what you will.

The big piece of research now would be to look back through 27 games to see exactly what we (and others) got those cards for and compare them to similar incidents in other matches……particularly those involving Leicester. We are, of course, one of the most fouled teams in the premiership so given the lack of protection we are afforded, I did consider that our players give the referee a lot of grief and show dissent for being persistently fouled. (That is actually how it feels watching the way our games are refereed.) This doesn’t stand up to superficial scrutiny though since we have been fouled exactly the same number of times as Leicester. From a previous bit of research I did though, I did notice that despite being fouled against more often than we commit fouls, we actually received more cards than the opposition. This could obviously account for our players showing greater frustration at refs. Having said that, this still suggests the ref is the cause of the problem. Without spending time analysing every single match and comparing cards per foul I can’t really say that this is a reason and without actually watching every match from start to finish I don’t think it would be possible to reach an informed decision. (I’ll be retired next year so maybe that’ll give me something to do!! Although I had intended to get a life rather than waste it!)

For me though, there are just far too many coincidences and/or anomolies in all these figures for this not to be considered a bit odd , in fact very odd.

A slight aside before I go. Since Tony et al and others who post on here have been drawing considerable attention to these statistics it appears to me that over the last month or so, Leicester and Liverpool have received a few more cards than they had previously whilst those who had been the most harshly treated have started to be treated slightly better. This may just be my perception but it’s certainly something I look at for……….I feel another research project coming on!

Nice work Mikey. Your third paragraph doesn’t actually say which team is treated the harshest.

As I have noted on many occasions, treatments are treated as a joke by PGMO (and largely the medja ignore them). It is possible for a player to go down without there not being some kind of acute contact immediately before going down. There are certain injuries which are a delayed response to an acute contact. Hence, just because a player goes down “on their own”, doesn’t mean that the treatment is not “inspired” by a foul.

You found some categorized discipline data. It would seem that treatments could also be categorized as to whether they are related to a foul or not.

There have been games where the player committing the foul, is the one who then requires treatment (self-inflicted).

All foul originated, not self-inflicted treatments should be associated with a free kick (which means the referee called the foul). I don’t believe the referee gives a free kick in most events of this type (more than 50%) of the time. Nor do I believe the referee gives a free kick a major amount of the time (more than 30%). I suspect a free kick is only given a minor amount of the time (more than 10%).

If treatments were given the same concern as discipline, there should be a correspondence between treatments and discipline. Or at least certain kinds of treatments. I don’t believe there is.

Treatments can be a place where players can waste time. I don’t believe I have seen any player be booked for simulation as a result of a treatment.

Many people seem to think treatments are trivial. If treatments were trivial, I would expect it to be rare for a player to require substitution after having received treatment. I believe last season there was one incident where a player requiring a treatment, was off the pitch for 10 or 11 minutes. Such long treatments and substitutions, are indicative of the seriousness of the injury.

By and large, PGMO ignore treatment situations.

Mikey I don’t have the time to go through the stats again, but maybe at the end of the season we ought to do that and see if there is a difference by then.

The notion that we could be influencing referee behaviour, consciously or sub-consciously, is really rather appealing.

One number Tony did not present (or if there, I missed it) was the average of tackles per foul. Id we add up all the tackles/foul across the 20 teams and then divide by 20, we get about 1.65. If I add up the square of the differences between this value and the tackles/foul across all teams, divide by 19 (the N-1 correction) and take the square root, I get the 0.19 standard deviation that Tony did mention.

One of the goals (IMHO) that Tony is embarking on, is to say that the EPL does not consist of a single population of data. Some teams may be favorite or not favorite, some officials may be favorite or not favorite, There might be geographic peculiarities, and so on.

In the above work, Tony specifically mentions Leicester a few times. So, it is likely that Tony thinks that Leicester is an outlier in the data.

There are many tests in the literature to try and detect an outlier in data. Chauvenet’s Criteria is perhaps the best known. A problem with Chauvenet’s Criteria, and most other tests for outlier, is that it technically can only find a single outlier in data. Once the data has been “cleaned” of this outlier (if the criteria suggested it was very likely to be an outlier); the data is NOT to be examined again for any other outliers.

An older test is Peirce’s Criterion.

https://en.wikipedia.org/wiki/Peirce's_criterion

This test can be used to find more than one outlier in data, which makes it a superior test in my mind.

Tony presents two ratios, tackles/foul and fouls/yellow. I suspect it Tony was to divide the total number of tackles by the total number of fouls, he would NOT find a ratio of 1.65. Nor would the total number of fouls divided by the total number of yellow cards be 6.04.

The total tackles/total fouls is a better estimator of central tendency, if the data for the EPL only comes from a single distribution of data. And likewise for total fouls/total yellows. Or, looking at the categories of yellows that Mikey ran into, the appropriate kind of yellows.

The reason is that there is probably a weighting factor needed to combine all these team datums.

Thanks tony for the article. I believe it was done in response to my request for clarification in an earlier article. I’m not a mathematician, and I hope Gord can clear this up. But your conclusion seems to be wrong. You seem to be saying since the SD for tackles/foul ratio is smaller than SD for foul to card ratio, that it means refs are calling fouls more consistently among the teams than how they are calling cards. This conclusion is wrong. From the figures shown it is fairly obvious the reason why the SD for F/C is bigger is because the numbers being analyzed are bigger (4.32-9.48 vs 1.29-2.08). If you used absolute fouls(200-294) and absolute cards figures(24-60) you would see that the SD for fouls called has a higher value than SD for cards issued. Based on your current reasoning, you would have to make the opposite conclusion than you have made in this article, from the same set of stats.meanwhile all that changed is the value of the figures you’re now working with. Like I said, I’m not a mathematician and so not an authority gord can have a look and educate us better.

I also want to say the article would have been easier for me to understand if the conclusion was plain. For example, “based on these figures it shows teams A,B,C have been hard done by the refs while team X,Y,Z seems to be having it easy”. thanks

Mikey

Great work. There must be hours of work in that little lot. Credit to you.

Still, having said that, I’m sure you’ll be hearing from a certain someone soon enough, telling you how paranoid you are and how you can MAKE statistics say whatever you want.

Well, if that is indeed the case, you couldn’t find a way to make the statistics say we’re the best team in the premier League and Liverpool are, despite appearances, actually crap, and in a relegation battle.

Tony,

It may be that the refs are noting “outside” influences, or it may be that they have to even the stats up so that when an end-of-season “analysis” is made by the media or somebody equally useless, nothing will stand out as obviously suspicious.

Referees even up stats in individual games.

Or rather, they often “almost” even up stats. And averaged over a season simple measure of what they do looks “even”.

Found floating very near the PIGMOB offices , was a few incompletely shredded pieces of some important looking document. These were then pasted together by members of a North London jigsaw club , who just luv this stuff .

And this is the satirical story I am going with !

And it reads as below…..

ADDENDUM TO LAW 12 –

(applicable only to Arsenal FC.)

-Any inappropriate or intimate handling of ball – includes fisting of said ball, or other suggestive actions.

-Giving the officals the evil eye(imagined or otherwise),or any suggestive eye up rolling, winking , squinting or any form spasmodic eye and facial tics.

-Inappropriate or suggestive snotting sounds ; throat clearing ; spitting, expectorating sputum or phlegm ;and expressing out nasal discharge .

– Any other signs, signals, gestures and hand movements that may be deemed to be offensive . These Johnny Foreigners types are very sneaky and sly.

Copies have been sent to the Arsenal for further action and education of the staff.