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**Statistics to win penalty shoot-outs**

With the World Cup upon us again we can perhaps look forward to yet another heroic defeat on penalties by England. England are in fact the worst country of any of the major footballing nations at taking penalties, having won only 1 out of 7 shoot-outs at the Euros and World Cup. In fact of the 35 penalties taken in shoot-outs England have missed 12 – which is a miss rate of over 30%. Germany by comparison have won 5 out of 7 – and have a miss rate of only 15%.

With the stakes in penalty shoot-outs so high there have been a number of studies to look at optimum strategies for players.

**Shoot left when ahead
**

One study published in Psychological Science looked at all the penalties taken in penalty shoot-outs in the World Cup since 1982. What they found was pretty incredible – goalkeepers have a subconscious bias for diving to the right when their team is behind.

As is clear from the graphic, this is not a small bias towards the right, but a very strong one. When their team is behind the goalkeeper apparently favours his (likely) strong side 71% of the time. The strikers’ shot meanwhile continues to be placed either left or right with roughly the same likelihood as in the other situations. So, this built in bias makes the goalkeeper much less likely to help his team recover from a losing position in a shoot-out.

**Shoot high**

Analysis by Prozone looking at the data from the World Cups and European Championships between 1998 and 2010 compiled the following graphics:

The first graphic above shows the part of the goal that scoring penalties were aimed at. With most strikers aiming bottom left and bottom right it’s no surprise to see that these were the most successful areas.

The second graphic which shows where penalties were saved shows a more complete picture – goalkeepers made nearly all their saves low down. A striker who has the skill and control to lift the ball high makes it very unlikely that the goalkeeper will save his shot.

The last graphic also shows the risk involved in shooting high. This data shows where all the missed penalties (which were off-target) were being aimed. Unsurprisingly strikers who were aiming down the middle of the goal managed to hit the target! Interestingly strikers aiming for the right corner (as the goalkeeper stands) were far more likely to drag their shot off target than those aiming for the left side. Perhaps this is to do with them being predominantly right footed and the angle of their shooting arc?

**Win the toss and go first**

The Prozone data also showed the importance of winning the coin toss – 75% of the teams who went first went on to win. Equally, missing the first penalty is disastrous to a team’s chances – they went on to lose 81% of the time. The statistics also show a huge psychological role as well. Players who needed to score to keep their teams in the competition only scored a miserable 14% of the time. It would be interesting to see how these statistics are replicated over a larger data set.

**Don’t dive**

A different study which looked at 286 penalties from both domestic leagues and international competitions found that goalkeepers are actually best advised to stay in the centre of the goal rather than diving to one side. This had quite a significant affect on their ability to save the penalties – increasing the likelihood from around 13% to 33%. So, why don’t more goalkeepers stay still? Well, again this might come down to psychology – a diving save looks more dramatic and showcases the goalkeeper’s skill more than standing stationary in the centre.

**So, why do England always lose on penalties?**

There are some interesting psychological studies which suggest that England suffer more than other teams because English players are inhibited by their high public status (in other words, there is more pressure on them to perform – and hence that pressure is harder to deal with). One such study noted that the best penalty takers are the ones who compose themselves prior to the penalty. England’s players start to run to the ball only 0.2 seconds after the referee has blown – making them much less composed than other teams.

However, I think you can put too much analysis on psychology – the answer is probably simpler – that other teams beat England because they have technically better players. English footballing culture revolves much less around technical skill than elsewhere in Europe and South America – and when it comes to the penalty shoot-outs this has a dramatic effect.

As we can see from the statistics, players who are technically gifted enough to lift their shots into the top corners give the goalkeepers virtually no chance of saving them. England’s less technically gifted players have to rely on hitting it hard and low to the corner – which gives the goalkeeper a much higher percentage chance of saving them.

**Test yourself**

You can test your penalty taking skills with this online game from the Open University – choose which players are best suited to the pressure, decide what advice they need and aim your shot in the best position.

If you liked this post you might also like:

Championship Wages Predict League Position? A look at how statistics can predict where teams finish in the league.

Premier League Wages Predict League Positions? A similar analysis of Premier League teams.

**Modelling more Chaos**

This post was inspired by Rachel Thomas’ Nrich article on the same topic. I’ll carry on the investigation suggested in the article. We’re going to explore chaotic behavior – where small changes to initial conditions lead to widely different outcomes. Chaotic behavior is what makes modelling (say) weather patterns so complex.

**f(x) = sin(x)**

This time let’s do the same with f(x) = sin(x).

**Starting value of x = 0.2**

**Starting value of x = 0.2001**

**Both graphs superimposed **

This time the graphs do not show any chaotic behavior over the first 40 iterations – a small difference in initial condition has made a negligible difference to the output. Even after 200 iterations we get the 2 values x = 0.104488151 and x = 0.104502319.

**f(x) = tan(x)**

Now this time with f(x) = tan(x).

**Starting value of x = 0.2**

**Starting value of x = 0.2001**

**Both graphs superimposed **

This time both graphs remained largely the same up until around the 38th data point – with large divergence after that. Let’s see what would happen over the next 50 iterations:

Therefore we can see that tan(x) is much more susceptible to small initial state changes than sin(x). This makes sense by considering the graphs of tan(x) and sin(x). Sin(x) remains bounded between -1 and 1, whereas tan(x) is unbounded with asymptotic behaviour as we approach pi/2.

**Modelling Chaos**

This post was inspired by Rachel Thomas’ Nrich article on the same topic. I’ll carry on the investigation suggested in the article. We’re going to explore chaotic behavior – where small changes to initial conditions lead to widely different outcomes. Chaotic behavior is what makes modelling (say) weather patterns so complex.

Let’s start as in the article with the function:

**f(x) = 4x(1-x)**

We can then start an iterative process where we choose an initial value, calculate f(x) and then use this answer to calculate a new f(x) etc. For example when I choose x = 0.2, f(0.2) = 0.64. I then use this value to find a new value f(0.64) = 0.9216. I used a spreadsheet to plot 40 iterations for the starting values of x = 0.2 and x = 0.2001. This generated the following spreadsheet (cut to show the first 10 terms):

I then imported this table into Desmos to map how the change in the starting value from 0.2 to 0.2001 affected the resultant graph.

**Starting value of x = 0.2**

**Starting value of x = 0.2001**

**Both graphs superimposed **

We can see that for the first 10 terms the graphs are virtually the same – but then we get a wild divergence, before the graphs seem to synchronize more closely again. One thing we notice is that the data is bounded between 0 and 1. Can we prove why this is?

If we start with a value of x such that:

0<x<1.

then when we plot f(x) = 4x – 4x^{2} we can see that the graph has a maximum at x = 1/2:

.

Therefore any starting value of x between 0 and 1 will also return a new value bounded between 0 and 1. Starting values of x > 1 and x < -1 will tend to negative infinity because x^{2} grows much more rapidly than x.

**f(x) = ax(1-x)**

Let’s now explore what happens as we change the value of a whilst keeping our initial starting values of x = 0.2 and x = 0.2001

a = 0.8

both graphs are superimposed but are identical at the scale we are using. We can see that both values are attracted to 0 (we can say that 0 is an **attractor** for our system).

a = 1.2

Again both graphs are superimposed but are identical at the scale we are using. We can see that both values are attracted to 1/6 (we can say that 1/6 is an **attractor** for our system).

In general, for f(x) = ax(1-x) with -1≤x≤1, the attractors are given by x = 0 and x = 1 – 1/a, but it depends on the starting conditions as to whether we will end up being attracted to this point.

**f(x) = 0.8x(1-x)**

So, let’s look at f(x) = 0.8x(1-x) for different starting values 1≤x≤1. Our attractors are given by x = 0 and x = 1 – 1/0.8 = -0.25.

When our initial value is x = 0 we remain at the point x = 0.

When our initial value is x = -0.25 we remain at the point x = -0.25.

When our initial value is x < -0.25 we tend to negative infinity.

When our initial value is -0.25 < x ≤ 1 we tend towards x = 0.

**Starting value of x = -0.249999:**

Therefore we can say that x = 0 is a **stable attractor**, initial values close to x = 0 will still tend to 0.

However x = -0.25 is a **fixed point** rather than a stable attractor**, **as

x = -0.250001 will tend to infinity very rapidly,

x = -0.25 stays at x = -0.25.

x = -0.249999 will tend towards 0.

Therefore there is a stable equilibria at x = 0 and an unstable equilibria at x = -0.25.

**The Folium of Descartes**

The folium of Descartes is a famous curve named after the French philosopher and mathematician Rene Descartes (pictured top right). As well as significant contributions to philosophy (“I think therefore I am”) he was also the father of modern geometry through the development of the x,y coordinate system of plotting algebraic curves. As such the Cartesian plane (as we call the x,y coordinate system) is named after him.

**Pascal and Descartes**

Descartes was studying what is now known as the folium of Descartes (folium coming from the Latin for leaf) in the first half of the 1600s. Prior to the invention of calculus, the ability to calculate the gradient at a given point was a real challenge. He placed a wager with Pierre de Fermat, a contemporary French mathematician (of Fermat’s Last Theorem fame) that Fermat would be unable to find the gradient of the curve – a challenge that Fermat took up and succeeded with.

**Calculus – implicit differentiation:**

Today, armed with calculus and the method of implicit differentiation, finding the gradient at a point for the folium of Descartes is more straightforward. The original Cartesian equation is:

which can be differentiated implicitly to give:

Therefore if we take (say) a =1 and the coordinate (1.5, 1.5) then we will have a gradient of -1.

**Parametric equations**

It’s sometimes easier to express a curve in a different way to the usual Cartesian equation. Two alternatives are polar coordinates and parametric coordinates. The parametric equations for the folium are given by:

In order to use parametric equations we simply choose a value of t (say t =1) and put this into both equations in order to arrive at a coordinate pair in the x,y plane. If we choose t = 1 and have set a = 1 as well then this gives:

x(1) = 3/2

y(1) = 3/2

therefore the point (1.5, 1.5) is on the curve.

You can read a lot more about famous curves and explore the maths behind them with the excellent “50 famous curves” from Bloomsburg University.

**Project Euler: Coding to Solve Maths Problems**

Project Euler, named after one of the greatest mathematicians of all time, has been designed to bring together the twin disciplines of mathematics and coding. Computers are now become ever more integral in the field of mathematics – and now creative coding can be a method of solving mathematics problems just as much as creative mathematics has always been.

The first problem on the Project Euler Page is as follows:

*If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.*

*Find the sum of all the multiples of 3 or 5 below 1000.*

This is a reasonably straight forward maths problem which we can solve using the summation of arithmetic sequences (I’ll solve it below!) but more interestingly is how a computer code can be written to solve this same problem. Given that I am something of a coding novice, I went to the Project Nayuki website which has an archive of solutions. Here is a slightly modified version of the solution given on Project Nayki, designed to run in JAVA:

The original file can be copied from here, I then pasted this into an online JAVA site jdoodle. The only modification necessary was to replace:

*public final class p001 implements EulerSolution* with *public class p001*

Then after hitting execute you get the following result:

i.e the solution is returned as 233,168. Amazing!

But before we get carried away, let’s check the answer using some more old fashioned maths. We can break down the question into simply trying to find the sum of multiples of 3 under 1000, the sum of the multiples of 5 under 1000 and then subtracting the multiples of 15 under 1000 (as these will have been double counted). i.e:

(3 + 6 + 9 + … 999) + (5 + 10 + 15 + … 995) – (15 + 30 + 45 + …990)

This gives:

S_333 = 333/2 (2(3)+ 332(3)) = 166,833

+

S_199 = 199/2 (2(5) + 198(5)) = 99, 500

–

S_66 = 66/2 (2(15) +65(15) = 33, 165.

166,833 + 99, 500 – 33, 165 = 233, 168 as required.

Now that we have seen that this works we can modify the original code. For example if we replace:

if (i % 3 == 0 || i % 3 == 0)

with

if (i % 5 == 0 || i % 7 == 0)

This will find the sum of all the multiples of 5 or 7 below 1000. Which returns the answer 156,361.

Replacing the same line with:

if (i % 5 == 0 || i % 7 == 0 || i % 3 == 0)

will find the sum of all the multiples of 3 or 5 or 7 below 1000, which returns the answer 271,066. To find this using the previous method we would have to do:

Sum of 3s + Sum of 5s – Sum of 15s + Sum of 7s – Sum of 21s – Sum 35s – Sum of 105s. Which starts to show why using a computer makes life easier.

This would be a nice addition to any investigation on Number Theory – or indeed a good project for anyone interested in Computer Science as a possible future career.

**Spotting Asset Bubbles**

Asset bubbles are formed when a service, product or company becomes massively over-valued only to crash, taking with it most of its investors’ money. There are many examples of asset bubbles in history – the Dutch tulip bulb mania and the South Sea bubble are two of the most famous historical examples. In the tulip mania bubble of 1636-37, the price of tulip bulbs became astronomically high – as people speculated that the rising prices would keep rising yet further. At its peak a single tulip bulb was changing hands for around 10 times the annual wage of a skilled artisan, before crashing to become virtually worthless.

More recent bubble include the Dotcom crash of the early 2000s – where investors piled in trying to spot in what ways the internet would revolutionise businesses. Huge numbers of internet companies tried to ride this wave by going public with share offerings. This led to massive overvaluation and a crash when investors realised that many of these companies were worthless. Pets.com is often given as an example of this exuberance – its stock collapsed from $11 to $0.19 in just 6 months, taking with it $300 million of venture capital.

Therefore spotting the next bubble is something which economists take very seriously. You want to spot the next bubble, but equally not to miss out on the next big thing – a difficult balancing act! The graph at the top of the page is given as a classic bubble. It contains all the key phases – an initial slow take-off, a steady increase as institutional investors like banks and hedge funds get involved, an exponential growth phase as the public get involved, followed by a crash and a return to its long term mean value.

**Comparing the Bitcoin graph to an asset bubble**

The above graph is charting the last year of Bitcoin growth. We can see several similarities – so let’s try and plot this on the same axis as the model. The orange dots represent data points for the initial model – and then I’ve fitted the Bitcoin graph over the top:

It’s not a bad fit – if this was going to follow the asset bubble model then it would be about to crash rapidly before returning to the long term mean of around $4000. Whether that happens or it continues to rise, you can guarantee that there will be thousands of economists and stock market analysts around the world doing this sort of analysis (albeit somewhat more sophisticated!) to decide whether Bitcoin really will become the future of money – or yet another example of an asset bubble to be studied in economics textbooks of the future.

**The Remarkable Dirac Delta Function**

This is a brief introduction to the Dirac Delta function – named after the legendary Nobel prize winning physicist Paul Dirac. Dirac was one of the founding fathers of the mathematics of quantum mechanics, and is widely regarded as one of the most influential physicists of the 20th Century. This topic is only recommended for students confident with the idea of limits and was inspired by a Quora post by Arsh Khan.

Dirac defined the delta function as having the following 2 properties:

The first property as defined above is that the delta function is 0 for all values of t, except for t = 0, when it is infinite.

The second property defined above is that the integral of the delta function – and the area of the graph between 2 points (either side of 0) is 1. We can take the bottom integral where we integrate from negative to positive infinity as this will be more useful later.

The delta function (technically not a function in a normal sense!) can be represented as the following limit:

Whilst this looks a little intimidating, it just means that we take the limit of the function as epsilon (ε) approaches 0. Given this definition of the delta function we can check that the 2 properties outlined above hold.

For the first limit above we set t not equal to 0. Then, because it is a continuous function when t is not equal to 0, we can effectively replace epsilon with 0 in the first limit above to get a limit of 0. In the second limit when t = 0 we get a limit of infinity. Therefore the first property holds.

To show that the second property holds, we start with the following integral identity from HL Calculus:

Hopefully this will look similar to the function we are interested in. Let’s play a little fast and loose with the mathematics and ignore the limit of the function and just consider the following integral:

Therefore (using the fact that the graph of arctanx has horizontal asymptotes at positive and negative pi/2 for the final part) :

So we have shown above that the integral of every function of this form will have an integral of 1, regardless of the value of epsilon, thus satisfying our second property.

**The use of the Dirac Function**

So far so good. But what is so remarkable about the Dirac function? Well, it allows objects to be described in terms of a single zero width (and infinitely high) spike, but despite having zero width, this spike still has an area of 1. This then allows the representation of elementary particles which have zero size but finite mass (and other finite properties such as charge) to be represented mathematically. With the area under the curve = 1 it can also be thought of in terms of a probability density function – i.e representing the quantum world in terms of probability wave functions.

**A graphical representation:**

This is easier to understand graphically. Say for example we choose a value epsilon (ε) and gradually make it smaller (i.e we find the limit as ε approaches 0). When ε = 5 we have the following:

When ε = 1 we have the following:

When ε = 0.1 we have the following:

When ε = 0.01 we have the following:

You can see that as ε approaches 0 we get a function which is close to 0 everywhere except for a spike at zero. The total area under the function remains at 1 for all ε.

Therefore we can represent the Dirac Delta function with the above graph. In it we have a point with zero width but with infinite height – and still with an area under the curve of 1!

**The Rise of Bitcoin**

Bitcoin is in the news again as it hits $10,000 a coin – the online crypto-currency has seen huge growth over the past 1 1/2 years, and there are now reports that hedge funds are now investing part of their portfolios in the currency. So let’s have a look at some regression techniques to predict the future price of the currency.

Here the graph has been inserted into Desmos and the scales aligned. 1 on the y axis corresponds to $1000 and 1 on the x axis corresponds to 6 months. 2013 is aligned with (0,0).

Next, I plot some points to fit the curve through.

Next, we use Desmos’ regression for y = ae^{bx}+d. This gives the line above with equation:

y = 5.10 x 10^{-7 }e^{1.67x }+ 0.432.

I included the vertical translation (d) because without it the graph didn’t fit the early data points well.

So, If I want to predict what the price will be in December 2019, I use x = 12

y = 5.10 x 10^{-7 }e^{1.67(12) }+ 0.432 = 258

and as my scale has 1 unit on the y axis equal to $1000, this is equal to $258,000.

So what does this show? Well it shows that Bitcoin is currently in a very steep exponential growth curve – which if sustained even over the next 12 months would result in astronomical returns. However we also know that exponential growth models are very poor at predicting long term trends – as they become unfeasibly large very quickly. The two most likely scenarios are:

- continued growth following a polynomial rather than exponential model
- a price crash

Predicting which of these 2 outcomes are most likely is probably best left to the experts! If you do choose to buy bitcoins you should be prepared for significant price fluctuations – which could be down as well as up. I’ll revisit this post in a few months and see what has happened.

If you are interested in some more of the maths behind Bitcoin, you can read about the method that is used to encrypt these currencies (a method called elliptical curve cryptography).

This is a quick example of how using Tracker software can generate a nice physics-related exploration. I took a spring, and attached it to a stand with a weight hanging from the end. I then took a video of the movement of the spring, and then uploaded this to Tracker.

**Height against time**

The first graph I generated was for the height of the spring against time. I started the graph when the spring was released from the low point. To be more accurate here you can calibrate the y axis scale with the actual distance. I left it with the default settings.

You can see we have a very good fit for a sine/cosine curve. This gives the approximate equation:

y = -65cos10.5(t-3.4) – 195

(remembering that the y axis scale is x 100).

This oscillating behavior is what we would expect from a spring system – in this case we have a period of around 0.6 seconds.

**Momentum against velocity**

For this graph I first set the mass as 0.3kg – which was the weight used – and plotted the y direction momentum against the y direction velocity. It then produces the above linear relationship, which has a gradient of around 0.3. Therefore we have the equation:

p = 0.3v

If we look at the theoretical equation linking momentum:

p = mv

(Where m = mass). We can see that we have almost perfectly replicated this theoretical equation.

**Height against velocity**

I generated this graph with the mass set to the default 1kg. It plots the y direction against the y component velocity. You can see from the this graph that the velocity is 0 when the spring is at the top and bottom of its cycle. We can then also see that it reaches its maximum velocity when halfway through its cycle. If we were to model this we could use an ellipse (remembering that both scales are x100 and using x for vy):

If we then wanted to develop this as an investigation, we could look at how changing the weight or the spring extension affected the results and look for some general conclusions for this. So there we go – a nice example of how tracker can quickly generate some nice personalised investigations!

**Cracking ISBN and Credit Card Codes**

ISBN codes are used on all books published worldwide. It’s a very powerful and useful code, because it has been designed so that if you enter the wrong ISBN code the computer will immediately know – so that you don’t end up with the wrong book. There is lots of information stored in this number. The first numbers tell you which country published it, the next the identity of the publisher, then the book reference.

**Here is how it works:**

Look at the 10 digit ISBN number. The first digit is 1 so do 1×1. The second digit is 9 so do 2×9. The third digit is 3 so do 3×3. We do this all the way until 10×3. We then add all the totals together. If we have a proper ISBN number then we can divide this final number by 11. If we have made a mistake we can’t. This is a very important branch of coding called error detection and error correction. We can use it to still interpret codes even if there have been errors made.

If we do this for the barcode above we should get 286. 286/11 = 26 so we have a genuine barcode.

**Check whether the following are ISBNs**

1) 0-13165332-6

2) 0-1392-4191-4

3) 07-028761-4

**Challenge (harder!) :**The following ISBN code has a number missing, what is it?

1) 0-13-1?9139-9

Answers in white text at the bottom, highlight to reveal!

Credit cards use a different algorithm – but one based on the same principle – that if someone enters a digit incorrectly the computer can immediately know that this credit card does not exist. This is obviously very important to prevent bank errors. The method is a little more complicated than for the ISBN code and is given below from computing site Hacktrix:

You can download a worksheet for this method here. Try and use this algorithm to validate which of the following 3 numbers are genuine credit cards:

1) 5184 8204 5526 6425

2) 5184 8204 5526 6427

3) 5184 8204 5526 6424

Answers in white text at the bottom, highlight to reveal!

ISBN:

1) Yes

2) Yes

3) No

1) 3 – using x as the missing number we end up with 5x + 7 = 0 mod 11. So 5x = 4 mod 11. When x = 3 this is solved.

Credit Card: The second one is genuine

If you liked this post you may also like:

NASA, Aliens and Binary Codes from the Stars – a discussion about how pictures can be transmitted across millions of miles using binary strings.

Cracking Codes Lesson – an example of 2 double period lessons on code breaking