## Monday, 21 September 2015

### The Third Law of Tottenham Hotspur.

A familiar sight for Spurs fans during the 2012/13 season was either Bale or Defoe peppering the goal with shots. The former hit the target on 70 occasions in the Premier League, scoring 20 goals and the latter was 40 for 11 goals.

For those who like their goals to come in a variety of guises, Bale was the one to watch, not only were his shots taken from a greater variety of distances than Defoe's, they also targeted a wider area of the goal.

A quarter of Bale's shots on target either entered the goal or required saving above the mid height of the net compared to Defoe, who virtually without exception aimed his shots low to the ground.

In terms of simply eye-balling the data, Bale had much more variety in his shot placement. If the goal were divided by high or low shots and then further by shots to the left, right or centre, giving six general shot placement areas in total, only one of Defoe's efforts in 2012/13 would have fallen into the "high" classification, compared to 19 for Bale.

 Bale, ponders whether to go high or low.
Often it is useful to have a numerical value to express something that is self evident, in this case Bale's wider variety of shot placement in relation to his then teammate, Defoe.

Entropy, as any chemist will tell you is the measure of the disorder in a thermodynamic system and it is increasingly used to describe the amount of disorder in a sporting context.

For example, how competitive a league is may be expressed by calculating how similar are the league points currently won by all sides in the division. (Hat Tip to The Times' Fink Tank)

In all its gory, degree level, physical chemical context, entropy or S invokes eigenvalues, Boltzmann constants, usually close to absolute zero, but in a simplified context of Bale verses Defoe, the equation p*ln(p) is sufficient to extract a usable figure from the data, where p is a probability or perhaps a proportion and ln is the natural logarithm.

Shot Placement, Bale and Defoe, 2012/13 Premier League.

 Player High Centre High Left High Right Low Centre Low Left Low Right Bale 12 4 3 21 16 14 Defoe 0 1 0 18 4 17

It's a simple matter to convert the six possible shot placements for each player to proportions, for example Bale's 21 low centred shots comprised 0.3 of his total on target shots. applying this to p*ln(p) gives us -0.361, we may discount the negative sign.

In total all of the six possible placements for Bale add up to 1.62. Had Bale's placement been randomly shared between all six possible areas, the sum of each p*ln(p) would have been 1.79, so Bale's figure to describe how varied his attempts were is 91% of an entirely random placement selection.

In contrast, Defoe by virtually shunning any shots which required a keeper to raise his hands above his waist, scored only 58% in 2012/13.

Not only was Defoe fixated in hitting his shots low compared to the more diverse Bale, he recorded the lowest figure for any player to have hit the target 25 times or more during 2012/13.

The sample sizes are small, but Balesque players included Suarez, Aguero, Dzeko, van Persie and Cazorla, while Lambert, Nolan, Kone and Sturridge were players in the Defoe camp.

Penalty kick placement is an obvious use for a method that quantifies variety of choice. Gerrard had more variety than Lampard in where he aimed his career spot kicks, although my Gerrard data is incomplete, so that conclusion may change.

And scores related to pass direction for individual players or teams may be used to characterize team styles.