Why Sky Sports News is wrong about Cristiano Ronaldo

In this article here- SKY SPORTS

Mr Balague makes a pretty hasty claim that Ronaldo's days as a goal-scoring machine are behind him. In this article we'll take a closer look at that claim and see if it stacks up to mathematical scrutiny. This is an actual piece of maths I did with a student recently in a statistics lesson. Ok, so first of all, let's get the numbers. To simplify matters we'll just look at league goals over his entire career:

Which if you graph looks like this:

Pretty decent. Now we can see that from 2010 up until 2015, he's been pretty consistent in his goal scoring capabilities. Let's take that then to be the mean rate of goals, which works out to roughly 1.164 goals per game. This is a Poisson distribution, with mean rate of 1.164. We take this to be the null hypothesis. The alternative hypothesis then is that the mean is less than that. Let's do a standard 5% tolerance test. Now let's look at the most recent season gone. In this he's played 28 league games, which means if he was scoring at the mean rate that would be roughly 32 goals we would expect. So to test this we would need to establish what the probability is of him scoring a measly 27 goals is when his supposed average is 32. In mathematical language then, what is P(X <= 27) when X~Po(32).

Since the Poisson tables don't go up to 32, we approximate using a Normal distribution N(32, 32). So, to look this up in the tables we must translate it to the standard Normal distribution, so z = (27 - 32 ) / 32 = -0.16 roughly. Looking that up in the Normal tables gives a probability of 56%. But remember the tables only give us the value for the positive side. As in the diagram below we've essentially worked out the probability it is less than +0.16, which is the orange region, but we want the probability it is less than -0.16, which would be the blue region.

Hopefully it's easy to see that they both add up to one, so we must take the result to be 44%. So what have we established? Well according to the null hypothesis, we would have a 44% chance of obtaining the goal scoring record we have seen of Ronaldo this season. This is well outside the very reasonable 5% tolerance band we allowed ourselves. So under these assumptions, we are forced to conclude that we do not have enough evidence to support Mr. Balague's claim that Ronaldo is past his best. I'm quite disappointed to see such a lack of mathematical rigor in our sports analysis. In the future I'd like to see football pundits backing up their wild claims with some sound statistical analysis, rather than (what has now been made clear) is just his opinion and nothing else. With role models like these football pundits, it's no wonder mathematical standards in schools are dropping at such an alarming rate.

(Published on 15 Mar 2016)

Quality not Quantity

There is a lot of focus in Economic theory on the importance of GDP. GDP is short for "Gross Domestic Product", gross meaning total before any deductions (like tax), domestic meaning that we are just looking at the country in question, and product meaning how much was produced. So in a nutshell it means how much stuff was made and sold and how many services were provided by a country in a given time period. There's a pretty big problem with this which I think is being overlooked. Let's take a very simple example to make my point. Imagine a desert island, something like this:

that can make exactly 3 products. One is coconuts, the other is fish, and the third is a hallucinogenic cactus. Let's say in 2013 the output per year was as follows:

  • Coconuts - 2000
  • Fish - 1000
  • Cactus - 200
Then say in 2014 the output was this:
  • Coconuts - 2500
  • Fish - 2000
  • Cactus - 1000
As we can see, production has gone up significantly, and therefore GDP has increased.

The basic idea with GDP is that bigger is better, which makes sense on first glance. With more coconuts available, they become cheaper so people can either buy more, or have some extra money left over to get an extra fish. But with fish production up as well, they are fortunate enough to have more left over cash for the cactus. By having more goods and services produced, people are able to consume more and so they live better lives. It's a pretty decent approximation and if you have nothing else to go by, and it's not too unreasonable to just aim for "more" production. But I call into question the validity of this assumption because it ignores one important aspect, and that is the quality of the goods being produced. And by quality I don't mean how round the coconuts are, or how fishy the fish taste, I'm talking about what that product actually does for the person consuming it, and how it impacts their life and their abilities. Eating a few extra coconuts isn't going to cause any harm, in fact it might be beneficial to be eating a bit more food so you have enough energy to do all the things that living requires. There is a problem though with the wild excesses of 2014, let's look at the production figures for 2015:

  • Coconuts - 500
  • Fish - 200
  • Cactus - 1200
Oh dear, what has happened?

It seems that the population of the desert island has become addicted to the psychedelic cactus. They have diverted all their resources into producing more cactus, and since they are whacked out of their minds half the day are barely finding the time to fish and tend to the coconut plantations. Production has clearly suffered. In fact the bumper produce of 2014 has been a direct cause of the catastrophic drop in GDP in 2015. So is it really true that just more production is better for the economy? And higher GDP with no regard to the real value of the goods being produced means increased growth and prosperity? Clearly there are many goods that benefit us, make us more able and lead to increased productivity and welfare for the citizens who consume those products. I'm able to run my tutoring business much more effectively and easily with a laptop than I would be able to without one. My car is also enormously beneficial compared to riding a horse to lessons as I probably would do without a car.

However, could there be products that are actually harming the economy in the long run? Things that through decades of consumption are producing less able and less productive workers? Then the extra GDP we are celebrating now is going to lead to reduced or stagnant GDP in the future? I'm not going to start pointing fingers here, I'm more brainstorming than anything. But what about Netflix? I mean, doesn't that make people do less stuff? They sit around more watching highly entertaining videos, rather than say learning a new language, or an instrument, or whatever people did before we had 24/7 on-demand media. It takes a lot more self-control and discipline to learn an instrument than it does to watch Breaking Bad, aren't those skills useful in the workplace as well? When I was younger, sometimes I'd just sit in my room and open my science textbook and look at all the cool pictures. Then I'd find myself even reading it sometimes!! Then in class I sometimes had a good idea of what the teacher was talking about cos I'd already read about it!!!

It's almost like me spending that spare time I had on learning improved my education and prospects in life. I'm pretty much 100% sure if I had had an Xbox with on-line gaming and Netflix when I was a boy I'd have been cramming in as much gaming and TV shows as I could, I mean I was already pretty bad as it was. I would eventually get bored of playing video games because there's only so many times you can do the same thing, and the Simpson's was only on once a day and I didn't like Hollyoaks so that meant I had time in between those things to do other things. Also doesn't really unhealthy addictive food create people that are more prone to disease and will miss work more often? All the economists are fist-pumping the air when GDP is through the roof, but if that's being driven by alcohol manufacturers and fast-food joints, there'll be articles in 15 years time trying to analyse why productivity is down and the governments tax breaks don't seem to be stimulating the economy. Sick people can't work and contribute to the economy, so how is it a good thing for these types of companies to be making large profits now and "growing" the economy when they in reality are shrinking it in the long run?

(Published on 1 Feb 2016)

Causation and Knowledge

So far I've been able to identify four key different areas of learning and understanding. The most primitive form of learning, is just factual. Just knowing facts. Like you know your name, you know which country you live in. Facts are important to some extent, because they set the context for everything else. They are limited though and we cannot consider memorising hundreds of facts to be a good standard of education.

The second of these is what I'd call "black box" learning. This is something we do every day, it's things that we know but we don't know how or why, we just know. Like many people know how to use a computer, they do not know how a computer works. They know how to drive a car, they do not know how a car works. The inner workings of the object in question are completely hidden and unknown, as if they were sealed within a black box. Again, this is a valid form of learning, as we do not all have the time to learn everything in great detail and do sometimes need to use things that we do not fully understand. We know what we need to know.

The third type of learning, I'd call just simply understanding. Like you're told how and why stuff works and what's going on "under the hood". Many people who are at this level in their field are considered experts or professionals. Take for example a good electrician. He not only knows how to wire a house, he knows the importance of the various components in the circuit, and he knows what he can do without and what is an absolute necessity. With this level of understanding, you have enough knowledge to tinker and adapt the "recipe" of what you're doing. You can predict the likely outcomes of any changes made, you can create and improve or adapt depending on the situation. I think this is important, as it allows you to be the one that makes decisions, to be the one that causes things to happen, not be always dancing to someone else's tune.

The fourth level of knowledge, I'd say comes from actually creating knowledge. Being the person that comes up with the solution or the method. Not to say that none of the other three are less important than this one, because they all have their parts to play in society, but this one can become important when a new unsolved problem arises. Before we had mains electricity, there was no textbook with the answer at the back, somebody had to figure it all out for themselves. In reality I think the cycle of knowledge goes in the reverse order to what I put here. Somebody poses and solves a problem, they develop a method of reliably handling the problem, then they teach other people to understand it. People that don't have the time or need to understand it, but do need to use it, simply "know" it on a black box level. Scientists developed or created the ideas of mains electricity, electricians and engineers understand those ideas, the rest of us simply know how to flick a light switch.

I think one of the main problems we have in the UK state education system, is that learning is largely restricted to the first two levels, of simply recollecting facts and memorising and drilling methods. Applying a known method to an already studied type of problem is called "problem solving", which I feel stretches the definition a bit. It's more of a memory test. I think the brighter students naturally develop some of the level three actual understanding, and they really do know what they are doing and why, but I think this is the minority. I know this because I regularly teach students that barely even approach this level, and interestingly enough some of them are able to get good marks on the exam without even touching it. And I think that's precisely the cause of this, mainly that teachers are under so much time pressure to get students up to scratch, that they mindlessly drill them on methods and methods until the students can reliably reproduce them in an exam.

What's quite sad is the fourth level has been completely stripped out of the national curriculum, and I think the reason is very simple - it's extremely hard to test. In fairness, it's not that easy to teach, but it is very hard to examine. But by removing it altogether, by shirking away from that challenge, we're training a nation of people only able to follow instruction. People who, for the most part, don't really understand what they are doing or why, they simply know that they're supposed to do it. The great advances in civilisation are left to chance, the shining stars among us who, for whatever reason, are capable of going against the grain. With some of the challenges we face as a society, neglecting this problem is a severe risk to take.

(Published on 1 Feb 2016)

The Responsibilities of Students

I wanted to write an article for students so they really understand what they need to be doing. It is a guarantee that if you don't even know the things you're supposed to be doing, you're going to fail to do them.

1) Take care of yourself

I think a lot of people are unaware of the link between their physical condition and mental performance. This is something you should learn about for yourself, but I will provide the basics here. First of all, drugs of any kind, recreational or those prescribed by a psychologist, will adversely affect your ability to study, learn and remember things. They can also interfere with your emotional stability. Alcohol is also something that can inhibit your ability to study and is something I recommend you take in moderation if at all. You should not take it before a school day or around exam times.

Another thing to watch out for is excess sugar consumption, eating food laden with artificial flavourings and preservatives, and highly processed junk food. Again, these are things you should eat in moderation. Making sure you get a good balanced diet will mean you sleep better, can concentrate more and have more energy. It will help you increase your grades. I know an extremely good nutritional advisor, contact me at richardhp at gmail dot com and I can put you in touch with her.

2) NEVER go past something you don't understand

Don't ever say to yourself that "you'll figure it out later". You are setting yourself up for failure as you will then struggle to understand all the following topics that build on that one. If you are taught something that isn't clear, you need to work that same day to understand it before the next lesson. Go over your notes, research it on the internet, read your text book, go back and ask your teacher, your friends, or book yourself a tutor! There is no excuse for this as there are many ways of finding out for yourself. Avoiding it is study suicide.

3) Be thorough.

As the saying goes, the devil is in the detail. It is the small differences in the details that make all the... difference! Your exam papers will be testing your understanding of the minor details, so it's not enough to just get the basic idea and then hope to wing it from there. That is certainly a good start, but you will not score highly on your exams if you cannot grasp the finer details of the subject.

4) Be interested

As a maths tutor, I often get asked "what is the point of studying blah (insert one of: algebra, trigonometry, geometry etc.)? Well in my opinion there are lots of reasons to study it, but here is one that beats them all - because it's interesting. When I was at school I hated Art and English and was only interested in the sciences. Now I am able to be interested in those things and more, and I really regret not putting more effort into it. I have met a great number of adults who tell me they hated maths at school but now wish they'd understood it better.

All of the things you study at school have been studied by hundreds of the smartest people in the history of the Earth over centuries and millennia. Isn't there a reason for that? You can make anything interesting, and when you understand things, they are interesting. Don't use this one as an excuse to cover up that you won't put the effort in to figuring something out. You will probably regret it when you're older.

5) Be in the driving seat

I think this one is the most important one for several reasons, which I'll explain. but what I mean by this is it should be the student that is searching for understanding themselves, not sitting back and waiting to be taught. You have a whole world of information to study and learn about, and a free education system where there are trained professionals whose full time job is to help you learn it! It is a fantastic opportunity and one that does not exist when you are older than 21 usually. When you're older, if you want to learn something, you gotta pay someone!

You need to be looking for yourself, checking things out, looking at ideas and thinking about them, discussing them with your friends or teachers. Imagine you have a theme park to explore, do you grab the map and run around looking for the best rides? Or do you take the guided tour? I promise you will learn the most by taking your own initiative and pushing yourself through your studies.

The second reason this is important is that is how life works basically. Being able to drive yourself forward and figure things out is the main skill required by all companies. If you want a good job, you need to be able to do this, so start practising now! No one wants to hire someone who just sits around waiting to be told what to do. It's exhausting having to run someone and always give them their next task. Employers want someone who will come in, figure out what the job needs and will get on and do it. Start now.


Well there you have it, that's what you need to be doing. If you're having trouble with your studies take a look and see if it's down to any of the points here. If you think it isn't , then I'd love to hear from you. Do get in touch and tell me what the problem is and I'll see if I can help and put the solution onto this page!

(Published on 3 Mar 2015)