## Shot Stopping vs Preventing Shots Revisited

In a previous post I suggested that for a goalkeeper shot stopping is more valuable than preventing shots. By using some simple maths I showed that at real world save% levels (c. 60-80%) increasing save% by 1% is 1.5x to 4x more valuable than decreasing SoT/match by 1%. In this post I will build on that by looking at how often goalkeepers come to catch or punch crosses and how valuable those aerial contributions are using data from the first 16 rounds of the 2010/11 Bundesliga season. [1]

Goalkeepers face a lot of crosses every match. To be more specific, based on the data a goalkeeper faced on average 20.5 crosses per 90 minutes, or roughly one cross every four and a half minutes. Despite the high volume of crosses coming their way the average goalkeeper is passive in dealing with them, making only 1.24 catches and 0.83 punches per 90 minutes on average. Of all the crosses they face goalkeepers catch or punch only 10.1%. [2]

Of course some goalkeepers are more active in the air than others. The most active goalkeepers dealt with c. 15% of total crosses – Weidenfeller (15.2%) and Neuer (14.9%) – while the least active dealt with c. 6% – Sippel (6.4%) and Mondragon (6.2%). If we look specifically at the most dangerous crosses, the crosses into the six yard area, the average goalkeeper deals with only 32.8% of them and even Manuel Neuer, arguably the strongest goalkeeper in the world when it comes to aerial ability, deals with only 52.5%.

The relevant point is that even the most active goalkeepers deal with only a small minority of crosses. In addition to the low volume of aerial contributions there is also the question of how valuable each contribution actually is.

Let’s look at it from another angle and ask how valuable is a cross on average? If we simplify the question a little we could say that the value of a cross depends on three things:

1. What percentage of the time the cross finds a teammate

2. What percentage of the time the teammate is able to convert the cross into a shot on target

3. What percentage of the time the shot on target results in a goal

Intuitively it seems clear the value of the average cross is very low. The three-way parlay of finding a teammate, getting a shot on target and the shot resulting in a goal is very unlikely. According to a stat from @Orbinho via @11tegen11 in the Premier League the average cross leads to a goal only 1.6% of the time, so the value of a cross is 0.016 goals.

If the value of a cross is so low then the value of a goalkeeper dealing with that cross must also be low. If the average goalkeeper claims two crosses per 90 minutes and the most active goalkeepers claim just under three (Neuer 2.75), then the value of being elite at claiming crosses is c. 0.015 goals per match. Contrast this to shot stopping were the value of being an elite shot stopper is c. 0.2-0.3 goals per match.

In summary, crosses are a high volume low value action. Even the most active goalkeepers deal with only a small minority of total crosses, and each contribution has only a small value. The value of preventing shots by dealing with crosses pales in comparison to the value of shot stopping.

—

Something I didn’t address is the psychological benefit of having a goalkeeper who is good in the air on your team. I can’t speak from the perspective of a high level player but as a fan I know having a goalkeeper who is poor in the air fills me with fear and worry, and having a goalkeeper with a strong aerial presence has some positive value beyond simply the value they create by dealing with crosses. However I find it very difficult to believe that value could ever be enough that I would rather have an elite aerial contributor than an elite shot stopper.

I would also add that my view on the psychological benefit is that it comes not so much from dealing with a lot of crosses but more from simply not making any mistakes when you do deal with them. There are few things more frightening in football then seeing your goalkeeper flap at a cross, and that can have a negative effect on a team’s confidence. I don’t think a goalkeeper who mostly stays on his line but when he does deal with crosses does so confidently would have anywhere near the same kind of negative effect.

—

[1] I would have used the first 17 rounds to get the full first half of the season but the data for the 17th round was unavailable. The sample is also missing the match between Mainz and Stuttgart from the first round due to incomplete data.

[2] Goalkeepers do also drop some crosses (roughly 1 in every 20 matches on average according to the data) and sometimes come for a cross and miss entirely which is not reflected in the data (some keepers more than others of course) which would increase the actual value by a few tenths of a percent.

## Shot Stopping and an Elite Defense

While reading a forum discussion on the topic of Manchester United’s next goalkeeper I came across an argument that went something like this (and I’m paraphrasing):

Manchester United have a world class defense that is able to limit the opposition to only a few shots per match and therefore it is not important that United’s goalkeeper is a world class shot stopper, it is far more important that he is good at organising his defense, claiming loose balls, catching and punching crosses and distributing the ball with throws and kicks.

To investigate whether shot stopping is less valuable for a team which concedes fewer shots I did some simple calculations:

x = number of shots on target faced

z = save percentage

y = goals concededTo determine goals conceded we can use the equation:

y = x – zx

If we increased z by 1%:

y(2) = x – 1.01zx

We can then combine the two and look at the percentage change in goals conceded, %y, and ask what happens to %y when we increase z by 1%?

%y = 100(((x – 1.01zx) – (x – zx))/(x – zx))

%y = 100((x – 1.01zx – x + zx)/(x – zx))

%y = 100(-0.01zx/(x – zx))

%y = -zx/(x – zx)

%y = -zx/x(1 – z)%y = -z/(1 – z)

So %y depends on z. At z = 0.6 a 1% increase in z leads to a 1.5% decrease in y whereas at z = 0.8 a 1% increase in z leads to a 4% decrease in y. The more relevant point is that %y is independent of x, in other words the value of increasing save percentage by 1% isn’t affected by how many shots on target you face.

The reverse of the original argument, that preventing shots becomes more valuable for a team that concedes fewer shots, isn’t true either. A 1% decrease in shots on target faced always leads to a 1% decrease in goals conceded.

%y = 100(((0.99x – 0.99zx) – (x – zx))/(x – zx))

%y = 100((-0.01x + 0.01zx)/(x – zx))

%y = (-x + zx)/(x – zx)

%y = -(x – zx)/(x – zx)%y = -1

Having an elite shot stopper is extremely valuable regardless of whether you have a world class defense or not.

## Shot Stopping vs Preventing Shots

When evaluating goalkeepers there any many factors you must take into account like shot stopping ability, aerial ability, command of the penalty area, communication, kicking ability, throwing ability, athleticism and many others. Creating an exact and reliable measure of each attribute is difficult but by simplifying the question somewhat we can at least get a taste of what is valuable and what isn’t. One way of doing this is to sort a goalkeeper’s defensive actions into two categories: shot stopping and preventing shots. The first is obvious enough. By the second I mean catching and punching crosses, dealing with through balls and other loose balls, organising his defense, and others, all things which help limit the number of shots the opposition takes. If we accept this rough divide we can then ask, what is the relative value of shot stopping and preventing shots?

Assume a goalkeeper faces five shots on target per match and saves 74% of them, thus conceding 1.3 goals per match. If we decrease SoT/match by 1% while keeping save% the same we get 1.287 goals conceded per match for a gain of 0.013 goals per match. If instead we increase save% by 1% (increasing to 74.74%, not 75%) while keeping SoT/match the same we get 1.263 goals conceded per match for a gain of 0.037 goals per match. So in this example increasing save% by 1% is almost three times more valuable than decreasing SoT/match by 1%.

How much more valuable increasing save% is depends on the save%. The take home point is that at any save% above 50% increasing save% by 1% is more valuable than decreasing SoT/match by 1%. At realistic top flight save% levels (c. 60-80%) it ranges from 1.5x to 4x more valuable.

Intuitively it makes sense that shot stopping is more valuable than preventing shots. When a goalkeeper makes a save he is preventing an almost certain goal, but when a goalkeeper comes to catch a cross or deal with a through ball he is preventing a situation where some percentage of the time the opposition will get off a shot, and some percentage of that time the shot will be on target, and some percentage of that time the shot will lead to a goal.

So while we are still some way away from an accurate and complete method of mathematically evaluating goalkeeper performance, a model which values shot stopping ability above everything else is very likely going to be the place to start.

—

One thing worth noting is that in this article I used the term save percentage or save% quite often. When I use the term I mean with it a goalkeeper’s real or true save percentage, not the save percentage statistic that you can sometimes find in the papers or on the internet. I think the save percentage statistic is a poor one because it is influenced by too many external factors which may have nothing to do with a goalkeeper’s real shot stopping ability. Things like how good the defenders and midfielders are, and more specifically things like how good the midfielders are at putting pressure on the opposition shooters and how good the defenders are at forcing opposition attackers into low percentage shooting sectors, as well as countless other factors, all influence a goalkeeper’s save percentage statistic even though they have nothing to do with how good of a shot stopper a goalkeeper actually is. Goals-to-games ratio which is used in Spain when they hand out the Ricardo Zamora Trophy to the best goalkeeper of the year and number of clean sheets kept which they use in England to give out the Golden Glove award suffer from the same problems.

We need better metrics. I don’t know what those metrics would be but something that takes into account shot quality and weights saves by degree of difficulty would be a decent first step. The point I’m trying to make (in a roundabout way) is that our inability to accurately measure a goalkeeper’s true shot stopping ability doesn’t mean shot stopping as an attribute is any less valuable. Shot stopping is extremely valuable, whether we can accurately measure it or not.

## The Value of a Red Card, Part 4

*This is the fourth and final part in a series of posts investigating the value of a red card.*

Knowing the value of a red card isn’t particularly interesting in and of it’s self. If we knew that the value of a red card in a certain situation was say -0.2 goals we could say that a player getting sent off then would be hurting his team significantly, but it’s not like we need to do a long calculation to know that being sent off is generally a bad idea. However we can apply the information to answering certain in-match questions, for example when is it correct to intentionally foul someone knowing you will get sent off if you do and what is the expected value (EV) of that foul.

Imagine the following situation: two average teams are playing on neutral ground with the score at 0-0 with M minutes left in the match. By using the information from part three of this series we could use this same process for a specific situation but for the sake of simplicity I will use the average case. The opposition striker has just beaten the offside trap and is through on goal in a clear goalscoring opportunity. You are the center back and again for the sake of simplicity let’s assume that you cannot win the ball off him fairly, your only options are to let him go and hope he doesn’t score or to foul him, give away a free kick or penalty and take the red card. Knowing what we know from the previous two articles we can estimate the value of letting him go and the value of fouling him to see which is the better option.

Let’s say in this particular example there are fifteen minutes left in the match and you are outside the penalty area so by fouling you are giving away a free kick from a dangerous position. How often would a team score from the resulting free kick? It seems unlikely to me that it would be more than 10%. What if you let him go and he is 1-on-1 with the goalkeeper? He is surely a favourite to score but he won’t score every time. Let’s assume he scores 80% of the time.

If you don’t foul: 80% of the time you lose a goal and continue for 15 minutes 11vs11 (the EGD is still zero as before) and 20% of the time he doesn’t score and you continue for 15 minutes 11vs11. Thus,

EV(no foul) = -0.8 goals

If you do foul: 10% of the time they score from the free kick but 90% of the time they don’t. Either way you have to play 15 minutes 10vs11 (which as mentioned in part two of this series is worth -0.27 goals). Thus,

EV(foul) = 0.1*(-1) + (-0.27)

EV(foul) = -0.37 goals

So in this example fouling is worth 0.43 goals in EGD.

It’s worth noting that even though we are calcuting this in terms of goals, at the end of the day what we really care about are league points, and there are situations where looking at a situation in terms of goals and in terms of points can give different results.

For example if instead of there being 15 minutes left in the match let’s say there is no time left and the referee will end the match as soon as the striker scores or misses from either open play or from the free kick after you foul him. Since there is no time left being down to ten men doesn’t have any effect so we can simply say that the EV of not fouling is -0.8 goals (the striker scores 80% of the time) and the EV of fouling is -0.1 goals (the team scores from the resulting free kick 10% of the time). So fouling is worth 0.7 goals. If the match is tied then the foul is also worth the same in league points, 0.7. But what if you are leading the match 1-0? In that case if you don’t foul the player 80% of the time he will score making it 1-1 and you will get 1 point from the match while 20% of the time he won’t and you will get 3 points for a total of 1.4 points. If you do foul the player 10% of the time you get 1 point and 90% of the time you get 3 points for a total of 2.8 points. So even though the goal value of a foul is 0.7 goals irregardless of whether the score is 0-0 or 1-0, the point value changes dramatically depending on the score.

We can derive formulas for the EV of fouling when the match is tied and when you lead by one goal.

If x is how often the opposing team scores from the free kick or penalty, y is how often they score from open play if you don’t foul and v is the value of the red card in points,

EV(foul|lead) = ((1x + 3(1-x)) + v) – (1y + 3(1-y))

EV(foul|lead) = 3 – 2x + v – 3 + 2y

EV(foul|lead) = 2y – 2x + vEV(foul|lead) = 0 when y = x – (v/2)

EV(foul|lead) > 0 when y > x – (v/2)

EV(foul|lead) < 0 when y < x – (v/2)EV(foul|tied) = ((0x + 1(1-x)) + v) – (0y + 1(1-y))

EV(foul|tied) = 1 – x + v – 1 + y

EV(foul|tied) = y – x + vEV(foul|tied) = 0 when y = x – v

EV(foul|tied) > 0 when y > x – v

EV(foul|tied) < 0 when y < x – v

So for example if you lead 1-0 in a situation where the value of a red card is -0.5 points and the opposition scores the resulting free kick 10% of the time EV(foul|lead) > 0 when y > 0.35, in other words if the opposition player scores from open play more than 35% of the time fouling is always going to be the better option. As another example if the match is tied in a situation where the value of a red card is -0.1 points and opposition scores the resulting penalty kick 75% of the time EV(foul|tied) < 0 when y < 0.85, in other words unless the opposition player scores from open play more often than 85% of the time letting him go will always be better than fouling. In an extreme case where the match is tied in a situation where the value of a red card is -1 point (for example right at the beginning of a match) and the opposition scores the resulting penalty kick 75% of the time EV(foul|tied) < 0 when y < 1.75, in other words letting him go is always correct even if he scores 100% of the time.

Remember that these formulas and calculations are only applicable in the specific case where the expected point differential between the two teams is zero. To apply them to real world scenarios we need to use the general form which is applicable in any scenario:

EV(foul) = (xn + (1-x)m + v) – (yn + (1-y)m)

EV(foul) = xn + m – xm + v – yn – m + ym

EV(foul) = y(m-n) + x(n-m) + vEV(foul) = 0 when y = x – (v/(m-n))

EV(foul) > 0 when y > x – (v/(m-n))

EV(foul) < 0 when y < x – (v/(m-n))

where m is the team’s expected points if the opposition doesn’t score and n is the team’s expected points if the opposition does score. By using m = 3 and n = 1 we see that the equation is the same as EV(foul|lead) from before, and by using m = 1 and n = 0 the equation is the same as EV(foul|tied). Of course determining accurate m- and n-values in real world situations is difficult but the same basic method still applies.

These are just a few examples and you can play around with the different variables to try out different scenarios. Naturally these calculations suffer from the same limitations as the red card calculations in the previous articles and there are some things we didn’t take into account like the fact that by taking the red card the player also receives a one or three match suspension which has some negative value and that sometimes (certainly not often, but there is a nonzero chance) the referee will go easy on you and only give you a yellow card, but to again quote George E. P. Box, “essentially, all models are wrong, but some are useful.” I think this is useful.

## The Value of a Red Card, Part 3

*This is part three in a series of posts investigating the value of a red card.*

One of the problems with the work presented in part two of this series is that the calculations were done for the average case, two average Premier League teams facing off on neutral ground with both teams having an expected goal differential of zero, and as such it isn’t directly applicable to the real world. However we can improve on the results and approximate the solutions to real world problems by tinkering with the inputs.

If you recall, the original expected goal differential equation is:

Expected goal differential = ((goals scored per minute 11vs11 times the ratio of goals scored 10vs11 to 11vs11) – (goals conceded per minute 11vs11 times the ratio of goals conceded 10vs11 to 11vs11)) times the numbers of minutes left in the match

The average values for the different variables from the 03/04 to 09/10 Premier League seasons were

Goals scored per minute 11vs11: 0.0147

Ratio of goals scored 10vs11 to 11vs11: 0.72

Goals conceded per minute 11vs11: 0.0143

Ratio of goals conceded 10vs11 to 11vs11: 1.98

But what about in a specific case? Take for example Manchester United.

MU goals scored per minute 11vs11: 0.0215

MU goals conceded per minute 11vs11: 0.0081

As you would expect United were quite a bit better than average at both scoring and conceding goals. If we plug these numbers into the EGD equation, assuming for now the ratios of goals scored and conceded 10vs11 to 11vs11 stay the same, we can see that the EGD of United playing 10vs11 for 90 minutes is -0.05 goals, whereas for the average team it would be -1.62 goals. That’s if we assume the ratios of goals scored and conceded stay the same, but one could argue that United are better at playing with ten men than the average team and so the ratios should be adjusted as well. Say you think United are 10% better at scoring goals 10vs11 and 10% better at conceding goals 10vs11 so we change the ratios from 0.72 and 1.98 to 0.8 and 1.8. Using the same goals scored and conceded per minute numbers with the new ratios the EGD of United playing 10vs11 for 90 minutes would be 0.24 goals.

Now if you’re wondering how the value of a red card could be positive, remember that we aren’t calculating the value of a red card yet, we are calculating the expected goal differential of the team for the rest of the match. To determine the actual value of the red card we need to compare this figure with the team’s expected goal differential without the red card and the difference between the two is the value of the red card. I sort of glossed over this in the previous article because in the average case where the expected goal differential before the red card is zero it has no effect and the value of the red card is equal to the expected goal differential.

So what is the EGD of United in different 11vs11 situations? Well, I don’t know. If I did I would be making millions betting on sports and not writing silly articles for a blog no one reads, but we can guess.

Say United are at home against a bottom of the table side. It’s still scoreless and the second half has just begun when United have a player sent off. What is United’s EGD right before the sending off, after the sending off and what is the value of the red card?

On average over the sample United scored 0.0215 goals per minute but at home against a weak side that number should be higher, let’s say it’s 0.03 goals per minute. On average over the sample United conceded 0.0081 goals per minute but at home against a weak side that number should be lower, let’s say it’s 0.006 goals per minute. So United’s EGD(11vs11) is 0.024 goals per minute or 1.08 goals for the remaining 45 minutes. After the red card, using the ratios of 0.8 and 1.8, United’s EGD(10vs11) is (0.03*0.8 – 0.006*1.8) * 45 = 0.594 goals. The value of the red card is the difference between the two, or EGD(10vs11) – EGD(11vs11) = -0.486 goals.

We could also flip this example the other way and ask what is the value of the red card if the weaker away team receives it? Let’s say the bottom of the table team in this example is Wigan. On average over the sample Wigan scored 0.0111 and conceded 0.0160 goals per minute, but away at Old Trafford those numbers might be more like 0.0089 and 0.0192. Let’s also assume Wigan is worse than average at playing with ten men so instead of using the ratios of 0.72 and 1.98 we will use 0.6 and 2.2. Wigan’s EGD(11vs11) would then be -0.4635 goals and EGD(10vs11) would be -1.6605 goals. The value of the red card would be -1.197 goals. That a red card hurts a weaker team more than it does a stronger team shouldn’t come as much of a surprise.

Whether you agree with the exact numbers I used in the examples is irrelevant. You can play with different situations and different inputs all day long, what matters is the method and understanding how to adapt the equation to whatever scenario you want to investigate.

*In part four of this series I will use this method to investigate the issue of intentional red card fouling.*

## The Value of a Red Card, Part 2

*This is part two in a series of posts investigating the value of a red card.*

The purpose of this article is to try and quantify the value of a red card in terms of how big an impact it has on the expected goal differential of a team for the rest of the match. Unfortunately the value of a red card isn’t static, it changes depending on many different factors including how much time is left in the match, how good the team playing with eleven men is, how good the team playing with ten men is, how good the team playing with ten men is at playing with ten men and other factors. We can however estimate the value of a red card in an average case. To do so imagine two average Premier League teams playing on neutral ground (to offset home ground advantage). To simplify the calculations we need to make a few assumptions:

1. The probability of a team scoring in any given minute is constant, in other words the probability that a team scores in the first minute is the same as the probability that a team scores in the last minute. In real life this isn’t strictly true, for example in the 09/10 Premier League season roughly 44% of all goals were scored in the first half compared to 56% in the second, but if we don’t assume it to be true the calculation quickly becomes impossible.

2. Once a red card has been given there are no more red cards for the rest of the match. Again this isn’t true but it happens so rarely that the effect it has on the final result is negligible. To include various 9vs11, 10vs10 and 9vs10 scenarios into the calculation would only add another layer of complexity with no tangible benefit.

Now back to answering the question. First we need to know a few things:

1. How many goals per minute does the average team score when playing 11vs11?

2. How many goals per minute does the average team concede when playing 11vs11?

3. How does having to play 10vs11 affect how many goals per minute a team scores?

4. How does having to play 10vs11 affect how many goals per minute a team concedes?

Once we know all of the above we can set up a formula to determine the expected goal differential of the team playing 10vs11:

Expected goal differential = ((goals scored per minute 11vs11 times the ratio of goals scored 10vs11 to 11vs11) – (goals conceded per minute 11vs11 times the ratio of goals conceded 10vs11 to 11vs11)) times the numbers of minutes left in the match

To answer these questions I went through every match from the 03/04 to 09/10 Premier League seasons collecting data on how many minutes a team had to play 10vs11 and how many goals they scored and conceded in that time. I then used that information together with the number of goals scored in total over the sample to calculate the average goals scored by a team per minute 11vs11 and 10vs11 as well as the average goals conceded by a team per minute 11vs11 and 10vs11.

Overall the sample contained 380 red cards leading to 10094 minutes played 10vs11. According to the data the average goals

Scored per minute 11vs11: 0.0147

Scored per minute 10vs11: 0.0106

Conceded per minute 11vs11: 0.0143

Conceded per minute 10vs11: 0.0283

meaning the ratio of goals scored 10vs11 to 11vs11 was 0.72 and the ratio of goals conceded 10vs11 to 11vs11 was 1.98.

The goals conceded ratio seems fairly straightforward. According to the data teams concede roughly twice as many goals when playing 10vs11 than they would playing 11vs11, which seems reasonable. The goals scored ratio is slightly more interesting. In my first draft of the calculation, before I had collected any data, I guessed that a team playing 10vs11 would score roughly half as many goals as they would playing 11vs11. One friend suggested that the ratio should be even lower, closer to 0.2 or 0.3. I was somewhat surprised to see the data suggest a ratio as high as 0.72. This could mean that there is a flaw in the data, or it could mean that there is a disconnect between reality and our predetermined notions of how much playing 10vs11 actually affects a team’s goal scoring.

We can now plug these numbers into the expected goal differential equation from before and simplify it to get:

EGD = -0.018 * M

where M is the number of minutes remaining.

You can try different numbers yourself but for example in the extreme case of having to play 90 minutes 10vs11 the EGD would be -1.62 goals. With fifteen minutes left the EGD would be -0.27 goals. With five minutes left -0.09 goals.

To determine the value of the red card we need to calculate the difference between the EGD 10vs11 and the EGD 11vs11. Fortunately in the average case, since the EGD 11vs11 is zero for both teams, the value of the red card is equal to the EDG 10vs11 so we can use the above formula without having to worry about it.

Clearly this method isn’t perfect, but as a starting point I think it can be valuable. To quote George E. P. Box, “essentially, all models are wrong, but some are useful.” Hopefully this is useful.

*In part three of this series I will look at ways to tinker with the model to make it more real world applicable.*

## The Value of a Red Card, Part 1

*This is part one in a series of posts investigating the value of a red card.*

In an article published in the Journal of Quantitative Analysis in Sports entitled Estimating the Effect of the Red Card in Soccer: When to Commit an Offense in Exchange for Preventing a Goal Opportunity authors Jan Vecer, Frantisek Kopriva and Tomoyuki Ichiba analyse red card events from the 2006 World Cup and Euro 2008 and use in-match betting market data to attempt to quantify the value of a red card in terms of the effect it has on a team’s expected goal scoring rate.

Their sample contained 26 matches from the World Cup and 3 matches from Euro 2008, of which 2 were discarded because the red card was awarded at the very end of the match, for a total sample of 27 matches. The complete list of matches along with time and recipient of the red card, as well as before and after goal scoring rates for both teams are given in Table 1.

From this data the authors derive the goal scoring rate for the sanctioned team and goal scoring rate for the opposing team ratios. They found that the goal scoring ratio of the sanctioned team is 0.663 and the goal scoring ratio of the opposing team is 1.237. In other words when a team receives a red card they score roughly two-thirds as many goals as they would have scored had the match continued 11vs11 and the opposing team scores (or equivalently the sanctioned team concedes) roughly five-fourths as many goals as they would have scored (conceded) had the match continued 11vs11.

The method of using in-match betting market data is interesting and a useful alternative to methods used in previous studies (e.g. Ridder et al. (1994) and Bar-Eli et al. (2006)) attempting to quantify the value of a red card. Unfortunately for the authors the tiny sample size of red cards they use for their analysis makes any results they draw largely meaningless. 27 red card events is simply too small a sample size for us to put any weight in their findings. Before the results of this study can be trusted further work and a much larger sample size are needed.

*In part two of this series I will present my own work on the value of a red card based on data from past Premier League seasons.*