England 25 Wales 28

It seems churlish to distill down the most engrossing contest of the Rugby World Cup so far to a kicking contest, but so committed were the defence of both England and Wales that the try line was only breached once by each team and so kicking ultimately decided the fate of the first heavy weight encounter of rugby's initial experience of a "Group of Death".

Dan Biggar, a world class kicker with club and less often for country, save in the absence of Leigh Halfpenny was perfect from his seven penalty attempts and one conversion, while England's Owen Farrell was equally flawless from five penalties, a drop goal and a single conversion.

But the lingering controversy surrounds the penalty kick that never was from the right hand touch line, just outside the 22 which would have given Farrell the opportunity to kick the sides level with minutes remaining.

Instead England attempted to emulate, but failed to follow Japan's lead by kicking for the corner and the win, perhaps mindful of Wales' tendency to grasp defeat from glorious victory late in matches against the world's elite.

The generic chance of a kicker to succeed from where Farrell was required to kick the potential game tying penalty was around 60%, but Saracen's fly half raises the bar to just under 74% and until rugby stats become more widely available, we may only guess at how a points expectancy of 2.25 points from a touchline kick weighs up against a less likely haul of five or possibly seven points from a five metre lineout.

Ladies and gentlemen, your kicker for this evening.....Daniel Biggar.

Robshaw's decision to try to grasp the initiative in Group A partially eclipses the excellence of the kicks that were attempted and made on the night.

Both Biggar and Farrell were presented with a couple of difficult attempts mixed in with relatively straightforward tasks for kickers of their quality, but Biggar's extra opportunity, with the hindsight of knowing the failure of England's last driving maul makes Wales justifiable winners.

Simulating all kicks taken on Saturday, based around the position from where they were kicked on the pitch and the historical success rates of Farrell and Biggar, Wales win more points in 60% of simulations, with draws and wins for England roughly sharing the remaining 40% of outcomes.

However, if we throw into the mix Farrell's kick to the posts that wasn't called, England now win 41% of the simulations compared to Wales' 37% with 22% of the kicking simulations drawn.

And while context, such as the current score is inevitably missing from such an exercise, we can say that England did create the means to win Saturday's match slightly more often than they might have lost it, but they possibly chose not to use the fruits of their efforts to the best end......and Daniel Biggar's excellence did the rest.  

Additional research by @zanderk  :-)

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.