How can we know that strange “fouls per card” figures don’t come by chance?

This article follows from “If Premier League matches were fixed, how would we know?”

Data and analyses from Andrew, and from   Commentary from Tony


In the last article we saw that huge variations exist in the number of fouls it takes in games to get a card. But of course such figures as we saw in the last article could be produced by chance. “It all evens out in the end,” as is the cry of the eternal deniers of anything ever being wrong with referees.

So how are we going to see if this is likely or not?

The way we do this is to turn the thinking upside down and consider how clubs and/or referees in any league would use bookings as a way of doing one club down, while helping another club, while all the time leaving the door open for a compliant media carefully to ignore the issue and endlessly pretend nothing was amiss.

The figures below come from Footcharts web site except for the final column on the right which measures the difference between cards for and cards against which I have added.   So a positive number in the last column means that the club received more cards against itself than the opposition, while a negative number means that the club received fewer cards than its opponents game by game.

Team Pld Cards Cards per game Diff
For Agst For Agst
In the above a red card is equal to two yellows
1 Arsenal 20 52 46 2.60 2.30 6
2 Tottenham 20 51 40 2.55 2.00 11
3 Watford 20 47 37 2.35 1.85 10
4 West Ham 19 42 25 2.21 1.32 17
5 Man City 20 43 25 2.15 1.25 18
6 Bournemouth 20 43 41 2.15 2.05 2
7 Sheffield United 20 42 26 2.10 1.30 16
8 Man United 20 41 51 2.05 2.55 -10
9 Everton 20 39 41 1.95 2.05 -2
10 Wolves 20 38 47 1.90 2.35 -9
11 Norwich 20 38 38 1.90 1.90 0
12 Chelsea 20 37 45 1.85 2.25 -8
13 Aston Villa 20 37 56 1.85 2.80 -19
14 Burnley 20 36 20 1.80 1.00 16
15 Newcastle 20 35 32 1.75 1.60 3
16 Crystal Palace 20 33 49 1.65 2.45 -16
17 Brighton 20 32 33 1.60 1.65 -1
18 Southampton 20 30 40 1.50 2.00 -10
19 Liverpool 19 22 23 1.16 1.21 -1
20 Leicester 20 21 44 1.05 2.20 -23

So what do we learn from this?

The most obvious thing is that there is no immediately obvious pattern – and it can be argued that surely if there was anything in any of this search for data on fouling, there would be a pattern. And yet there is something here…

For we can see that high flying Leicester get many more cards against them, than for them.  However Manchester City near the top of the table are the reverse; they get more cards against them, than they get against their opponents.

As for Arsenal they get more cards against them than anyone else, and more cards against their opponents than even Villa and Palace.

That is enough of a jumble to make anyone looking at this think – “there is nothing wrong here – there is no pattern in the numbers.”  And at first I thought that,

Except, except, except….

What our figures showed in the first article was the enormous variation in the number of fouls it takes to give out a card.   So in our very first example at the start of the last article we saw that Newcastle could commit 12 fouls to get a yellow card while Arsenal committed seven fouls and got three cards.

Of course one can argue from this data that Leicester City, with their one card a game, are obviously an incredibly clean team that simply don’t foul, whereas their opponents try and chop them to bits.  And yet two seasons ago they received many more yellow cards than their opponents got in matches against them.  If that is so it has been an incredibly quick change – and one wonders why other clubs don’t copy their style.

Now I am not searching out these figures to prove that there is anything fishy in relation to refereeing in English football – as I have said many times, I think the thing that suggests that something is amiss is the extreme secrecy of PGMO, its refusal to allow referees to be interviewed, its offer of silence money to referees who agree not to say anything in public after they retire, its deliberate policy of restricting the number of referees, even though an increase in the number of referees would reduce the chances of Type III match-fixing considerably etc etc etc.

But what these figures do reveal are several unexpected issues, ranging from the number of fouls it takes to get a yellow card to Leicester’s paucity of yellows this season.

Combine this with the utter secrecy of the PGMO and its constant insistence of doing things differently from other leagues (refs not allowed to look at pitch side monitors, VAR brought in a year after most other leagues, very small number of referees etc) and yes, to me it does all look rather odd.

However even I was amazed at the variation in the number of fouls it takes to get a card, between Arsenal players and other clubs.   The fact that Arsenal get more cards per game, suggests Arsenal commit more fouls per game – but that is not the case.  It is down to the huge variance between the number of fouls it takes to get a card, in Arsenal matches.

And then there is Leicester who either have the cleanest team ever to walk on an EPL pitch, or they are being treated differently.  The average number of fouls per card in the Premier League this season is one card every 5.39 fouls.  Leicester have got 21 cards this season yet have committed 198 fouls – that is one card every 9.43 fouls.  They have to commit almost twice as many fouls as the average team, to get a card!!!

Now of course you might say that this is fair enough – obviously their fouls are of a very minor nature and not worthy of cards.  And unfortunately I can’t argue with that because the analyses that would allow such consideration are not available – although having watched Leicester (they are the nearest PL team to where I live and I have a number of Leicester supporting friends who will discuss such matters in an open and friendly manner) I am not at all convinced by this argument.

And they are happy to discuss this because we can make predictions of how matches will evolve based on these and other figures.  Quite simply find a bookie who will take your bet that Leicester’s opposition will get at least twice as many yellows as Leicester and you could be on a winner.  In fact using these figures I think we can start making quite a few predictions.

Which is not to say I am not alleging that refereeing is fixed in the the EPL.  My case has always been that there is every reason to be much more open about refereeing than is the case now, and the insistence on being secretive as PGMO is, and the insistence of mainstream media on not discussing any aspect of refereeing except when PGMO puts out its annual press release, merely makes some fans like me rather suspicious.

What I am saying is that the lack of yellow cards for teams such as Leicester (and here one might also add Southampton) is curious (not least given that Southampton has committed the sixth highest number of fouls and yet has the third lowest number of cards).

Roughly speaking, Arsenal and Leicester commit almost the same number of fouls but Arsenal get two and a half times as many cards. Of course you can say that Arsenal commit far worse fouls, but having spent half a day watch videos of both clubs while writing this, I think that one is hard to prove.





19 Replies to “How can we know that strange “fouls per card” figures don’t come by chance?”

  1. A clean team is largely composed of clean players. And likewise, a dirty team is largely composed of dirty players. What are the foul and card records for players at the teams they played for before they were at the team they are currently at? Now if some of those previous teams are in the EPL, we might even have data on what the same referee gives for fouls and cards to the same players, across different teams.

  2. Of course other factors also come into play. Take, for example, the two seasons Alan Shearer went without receiving one yellow card……..despite the fact that he committed more fouls than any other player in the premiership in both of those two seasons. Perhaps it helps when you are the captain and golden boy of the England football team.

    I have an idea which might help to identify some useful patterns which I might make next year’s study. I will look at a cards to fouls ratio in the premiership for clubs playing in European competition and then compare it to the how they fair in Europe. It may just be a tainted view that I have, but I do recall some English teams that were treated leniently in domestic football received far more cards in Europe whilst Arsenal were treated more leniently………or “fairly” as I prefer to call it!

  3. We saw on Sunday AMN getting a card for a similar foul that Jorginho did not get one for a few minutes earlier.

    Last week we saw Man$ity players committing numerous fouls for which no card was given.

    Either the refs are incompetent or biased.

    I cannot see a middle path.

  4. My preference is to start with the assumption, if it looks, swims and quacks like a duck……
    And these figures tell their own story as to what is going on, as does what we see game after game
    Wenger was silenced, Gazidis , well,looked to me at least took the path of staying out of things, perhaps his own ambition and agenda. Arteta preaches excellence and accountability, if this ref situation carries on, hope he is soon banging on Rauls door demanding he takes this on, the clubs survival in the EPL could even depend on it in the worst case. Arteta was calmly scathing of the Chelsea ref, and yes, he was a Chelsea ref on Sunday, for whatever reason.
    The club need to be a little less passive on this, can you imagine say Liverpool putting up with this ref shite?

  5. The data that is needed is simply obtained by the fraud squad. The banking system has some visibility but the income expenditure of the individuals involved can very quickly divulge criminal activity. Fraud does not need to be reported to be investigated.

    No matter how quiet the corrupt may be, the volume of their financial activity quickly deafens the ears of cheated. The BBC’s Panorama programme showed up some family members involved in unsavoury activity but fell silent soon afterwards. The billions involved in the game must be accounted for in order for the exchequer to take its fair share in taxes. One wonders how many whistle blowers does it need to find one off key?

  6. I’ve been banging on about our harsh cards per foul ratio for years, as I noticed it’s existence over 10 years ago.

    There has been the occasional season where we were treated average or even slightly better than, by we have been bottom or near bottom of these particular stats far too many times than would be expected, especially as we are supposedly a bunch of wimps that don’t like getting stuck in, or ‘in their faces’ as some commentators like to put it.

    Those stats stink.

  7. I think what would be helpful is to have an added column with the number of fouls per game for and agains Arsenal and compute that with the number of cards for and against Arsenal. This would give us a clearer picture of what is going on. Not that this comment has to be seen as a criticism of the post or an order… ;=)

  8. Walter
    We have had more than our fair share of really bad decisions against Chelsea over the years. Marcos Alonso on Hector Bellerin, Costa on Kos and Gabriel in the same match readily spring to mind and I am sure there are lots more by the likes of Cahill etc.

  9. There is also the issue of WHEN the bookings occur and to whom.

    For example putting the central defender on a card early on when he’s up against, say, Jamie ‘I cant stand up in a breeze’ Vardy, or a fullback when he’s up against Deli ‘Ooops there’s another divot’ Ali, can have a massive effect on the game.

    Also how many times players get booked for a first foul ?

    I thought I lip read David Luiz say to Pawson, ‘That was my first foul’ when he got booked.

    Which brings me to rotational fouling, because that clearly wont work if every player in turn is getting booked for their first foul, which given how many times we’ve received 5 cards or more in match, whilst comming relatively few fouls, suggests is happening.

  10. Having people point out that adding other data is wonderful, but what would be much better is if you went and got a copy of said data. Saving others of us time.

    Looking at some of the yellow card data for Martin Atkinson (not all of it), in particular this covers 156 games (some are not EPL). His average card rate is 3.48 cards per game. The variance in issuing cards is 4.11, which is slightly higher (might not be significant) than the mean (3.48).

    I’ll get back to typing in some more data on him.

  11. The negative binomial is the basic (simplest?) distribution which has a larger variance than a Poisson. If the rate “r” is integer, it is often called a Pascal distribution, and for floating point data it is often called the Polya distribution. The mechanism by which the variance is larger than a Poisson, is that successive counts are positively correlated (a higher than average observation is often followed by another that is larger than usual). Some of this is just regurgitation from Wikipedia.

    Wikipedia talks about the mean, variance and mode in terms of p and r. For this Atkinson data, p is about 0.153 and r is about 19.26. Which makes the mode about 3 (most common number of cards per game).

    0 7
    1 22
    2 25
    3 24
    4 37
    5 17
    6 11
    7 7
    8 4
    9 1
    10 1

    The observed mode is 4, which is close to the expected 3.

  12. @Gord,

    I did, computed what I found it but then Tony already had a 3rd post with the information.

    What I was not able to find was a downloadable stat on possession time, because I feel that this element ought to be available, indicating if a team is more on the offence then on the defence, which should have an effect on number of fouls.

    Interestingly, Tottenham have 4 fouls per card, Arsenal is second with 4.06
    We find City in 4th with 4.65, Chelsea is 9th with 5.19, Liverpool 18th with 7.41, Southampton 19th with 7.67 and Leicester last with 9.43

    Definitely there is something strange going on.
    I am not stats expert, so I’ll leave analysis to others…

  13. So in summary Tony, the 2 articles in this series so far say,
    1. We did an analysis of foul to card ratio in arsenal games and found a wide variation. But since we haven’t done the analysis for other teams games, we can’t say if this variation is peculiar to arsenal or not.
    2. We did a card for and against analysis and found arsenal to be mid table. We can’t make much of this table because teams at the top of the league table like Leicester and Man city were found to be on opposite spectrums of the for/against table, even Liverpool has only a slender for/against advantage while teams at the foot of the league like Southampton and Aston villa have a huge positive balance,so saying teams higher up are benefitting from favourable card balance would be far fetched.
    3. We only know that even though arsenal are midtable in the card balance (we get a little more than our opponents do) we get plenty of cards, the highest in the league (note our opponents do too). However we haven’t analysed the nature of the cards, the proportion that is due to fouls, the proportion due to petulance or retaliation, fighting etc. But whatever the case our opponents seem to get a lot too
    4. After seeing the above,we can’t find any objective evidence that something is amiss in the card balance, but based on the “evidence of our eyes” we know something is amiss. Why should Leicester have far lower card balance than us? We can’t say for sure that they play cleaner than us but we don’t think they do. We have a much better balance than man city, but we won’t talk about that, neither will we bother about the other teams we’re much better than or that Liverpool is mid table too.
    5. Okay, so we don’t have much of a case when the evidence is looked at, but why is pgmo so secretive and why do they employ so few refs?

    Tony I hope I was able to do justice to the articles with this summary?

  14. Hahahahaha, Tony, Lima is spot on can you tell us which of his points is off and why that is the case?

  15. Carrying on from the 2 posts (first is above).

    What is mentioned above, is 156 games most of which is Premier League. The data I am now starting to work is 662 games, over a much wider field.

    On 156 games, the average was 3.48 and the average was 4.11, leading to a value for p of 0.153 and r of 19.26, which predicts a mode of 3 (and we observed a mode of 4).

    On 662 games, the average has dropped to 3.24, with a variance of 3.48. The red card rate was 0.136 (variance of 0.148).

  16. Back again. I’ve hacked up the data about Atkinson.

    Over the 662 games, only 2 games are for the Community Shield, so on their own they won’t mean anything.

    There are 394 EPL games, for which we have a mean of 3.39 and a variance of 3.28 on yellow cards (possible under-dispersion).

    There are 61 FA Cup games, with a mean of 2.83 and a variance of 3.22 (over-dispersed) for yellow cards. Red card rate is about half of global or EPL rates.

    There were 22 League Cup games, with an average of 2.36 and a variance of 2.32. Smaller red card rate.

    His yellow card rate for Tier-2 (54 games) was 2.57, for Tier-3 (27 games) was 2.44, Tier-4 (24 games) was 2.5 and for Tier-5 (8 games) was 2.625.

    Atkinson is much happier throwing around yellow cards at UEFA games. For Champions League and Euro Champions games, he averages 4.27 cards per game (4.79 Var). For Europa League and UEFA Cup, he is throwing out 4.16 cards per game (5.71 Var).

    Over the years, every year since the 2002/3 season, he has a card rate of: 3.11, 2.35, 2.29, 2.05, 3.05, 2.60, 3.38, 3.90, 3.49, 3.45, 3.78, 3.03, 4.02, 3.74, 3.40, 3.58, 3.24 and 2.94.

    Time to see why Kolasinac already has a card in our game v ManUre.

  17. I am simply saying that if some unusual influence in games by the referees vis a vis fouls resulting in cards is going on, it would naturally be happening in ways that attempted to hide the influence, and the figures are consistent with such an hypothesis. It would clearly not happen overtly in each game all the time.
    The whole point of Type III match fixing which has been explored many times on this site, is that it happens when the opportunity arises for the ref, and these figures are consistent with such an approach. Probably Lima’s issues arose because he wrote his piece without actually reading the whole series of articles.
    I have said many times that this type of match fixing was only spotted in Italy in the end because of phone taps so the conversations behind the match fixing were heard. Clearly we can’t do that so all we can look for are figures which are unlikely to occur by chance or through consistency of application of the rules, and which would clearly benefit some teams and be to the detriment of others, and which are unlikely to occur by chance. I believe we have found these giving these strange results for Leicester, the results for the London clubs and so on.
    I guess the simple point being missed in correspondence is that if there is influence on matches it is going to be hidden and inconsistent – inconsistent because no one needs to fix a match if the required result is happening anyway.

Leave a Reply

Your email address will not be published. Required fields are marked *