## 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.

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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.