The maths behind penalties

Of the many areas. Penalties is another area I have looked at and I think the findings may be of use to any prospective England goalkeeper facing a penalty, football traders may also benefit!

Really you should score

First off, you can reach some easy conclusions. From spot to the goal line, the ball will travel just 12 yards. It is estimated that this will happen in about 0.30 seconds. Footballers should score from this distance, in fact, around 85% of penalties are scored from this distance. Around 6% are missed and the rest are saved!

Over time it appears more penalties are being saved. If you track things from the start of the English Premier League then it seems goalkeepers seem to be saving more.

The psychological side

Penalties tend to force a goalkeeper to make a decision about what to do, BEFORE, the ball is struck. He has to anticipate which way to dive. He could dive left, right or stand still. This is where a subtle piece of psychology comes into play. Perhaps you can call it the action / inaction bias.

Imagine the scenario, you know that any goal in a match in the knockout stages will be significant and that by saving a penalty you will almost certainly have a significant impact on the match and on your status. What do you do? If you stand in the middle of the goal like a lemon you could be a national pariah if a goal is conceded. Dive spectacularly and tip it around the post and the praise will come flooding in. What happens is that the goalkeeper ends up facing the same dilemma and regret pang as a gambler or trader. The goalkeeper is fearing the negative consequences of his actions, rather than making a logical and reasoned choice. This fear is so overwhelming it has a strange effect. In 286 matches analysed, the goalkeeper dived in one direction or another 268 times. He went for glory 94% of the time. Did the goalkeeper make a logical choice? To do this you need to analyse where the ball went.

Over the same 286 matches, the penalty taker more or less evenly distributed the shots left, right and centre either by design or accident. In the data I looked at there was a bias to the right-hand side but there could be many reasons for that. Excluding the bias, it was a random shot choice. Therefore the goalkeepers definitely were making a sub-optimal decision by choosing to dive so often. Penalty takers struck the ball down the middle nearly 29% of the time, goalkeepers only stood there around 6% of the time! If you are a goalkeeper, standing idly would appear to be a decent choice but as shots were equally distributed then any random choice would be sensible. If you are a striker then you should engage in some gamesmanship to encourage the goalkeeper to dive, then send the ball down the middle.

In the sample we were looking at, 32/268 penalties were ‘saved’, around 11%. Saved shots were reasonably equally distributed whether the taker sent them left right or down the middle. But, when you combine this with goalkeeping stats one thing really stands out. If the goalkeeper stands in the middle and a shot is sent in roughly that direction, it was saved 60% of the time. This is more than double the rate at which a goalkeeper who guesses right can save a kick. Obviously, if a shot is sent left and the goalkeeper goes right the strike rate is err.. zero.

Whichever way you look at it, standing still is actually a good tactic. It’s difficult to reach the ball in a far corner from a good penalty taker, so it’s better to capitalise on nervous takers who don’t want to risk missing. Of course, this all depends on one penalty being independent of another. Let on that you are going to stand still and you increase the chance that the ball will not go down the centre. That is where it gets more complex, should your strikers assume the goalkeeper will dive or stand still? Logic tells you to send it into the corner but reality indicates that at least half the time the goalkeeper will dive that way. It all comes down to psychology.

There are a number of other stats you will be interested in rather than type them all out, they appear on the video at the top of this article.

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