WHEN the World Cup, now under way in Russia, progresses to the knockout phases of the competition on June 30th attention will focus on the dreaded penalty shoot-out. Forty years ago, if a game was level after 120 minutes, the winner was decided by luck: a simple coin-flip. But in 1978 the rules were changed to create results that, at least in some sense, depend on skill. The question is, how much skill? Since 1982, the first competition in which penalty shoot-outs actually happened, there have been 26 of them—with seven of the 18 teams in the nine pertinent finals having arrived there thanks to success at penalties, and two of the finals themselves having been decided by them.

The format of a shoot-out is simple. Teams take it in turn to try to kick five penalties past the opposing team’s keeper into the goal. If the score is even after five penalties a side then “sudden death” ensues: victory is achieved by a single winning kick that is not successfully replied to. Whether this is truly less dependent on luck is moot. Analysis suggests that no relationship exists between a team’s general quality and its success in such shoot-outs. What analysis does suggest, though, is ways to improve the odds of victory.

The first is to go first, if given the option. That option is, admittedly, dependent on the toss of a coin. But if you win the coin toss you should take it, according to Ignacio Palacios-Huerta of the London School of Economics. After analysing data on 1,000 penalty shoot-outs in the World Cup and other competitions, Dr Palacios-Huerta found that teams which kick first win 60% of the time. Moreover, toss-winning captains do usually take this option, so FIFA, world football’s governing body, is trying out a system similar to a tiebreak in tennis, in which teams A and B take turns to shoot first: AB then BA then AB and so on. The current World Cup, however, will keep the AB then AB format.

The toss having been won or lost, the teams decide the order in which players will take their kicks. Coaches typically select the best players to kick first, leaving the worst until last. Kickers are successful three-quarters of the time, on average, according to an analysis of penalties by The Economist. Yet the success rate falls by 12 percentage points for the fourth of the five pre-sudden-death penalties. This is where first-mover advantage appears to matter. The success rate in the fourth penalty for the team shooting first is 70%, whereas for the team shooting second it is just 56%. Thorough analysis of player sequencing by Dr Palacios-Huerta suggests that the importance of the five penalties is U-shaped: the first and fifth matter most; the third, least. So the best penalty takers, either in technique or those who can cope with stress, should be selected with that in mind.

Once the sequence of kickers is settled the ball is placed on the spot, 11 metres (36 feet) from the goal, the mouth of which is 2.4 metres high and 7.3 metres wide. A well-struck ball arrives at the goal line in just half a second, meaning that the goalkeeper must dive pre-emptively in the direction that he expects the kicker to shoot. Goalkeepers find high balls the hardest to deal with—just 3% of penalties aimed halfway up the goal or more are saved. Yet there is a tendency for these shots to miss the target: 18% of high shots do so, as opposed to 5% of low shots. Overall, though, allowing for misses and saves, high shots are successful 79% of the time compared with 72% for low shots (see chart).

As to the direction, left, right or centre, of both the kicker’s shot and the goalkeeper’s pre-emptive dive, it is best to be as unpredictable as possible. The data suggest there is little difference in success rates between shots that are aimed left, right or down the middle. Yet it is easier for a right-footed player to give the ball speed by aiming towards what is, from his point of view, the left-hand side of the goal (the keeper’s right), and vice versa for left-footed players. On average, kickers strike the ball in this more natural direction 25% more frequently than in the other direction. Goalkeepers know these preferences and dive in those directions in matching proportions, in an attempt to exploit this bias.

Preparation helps, too. The Netherlands substituted in a specialist penalty stopper, Tim Krul, just ahead of their shoot-out with Costa Rica in the 2014 World Cup. It worked. He dived in the correct direction all five times and saved two penalties. Conversely, there is no substitute for kicking accuracy. Germany, with an 86% penalty success rate, has the best record of any top international team. England’s record, by contrast, is a dismal 66%.