Ep 95: Merry Christmas from Mr Science!

Reindeer and Santa Claus are the topic of this week's Mr Science Show. With Christmas here, we thought we'd look at some Christmas news, and this week we take a look at reindeer facts and the problems Santa is having at the North Pole. Due to global warming, and the global financial crisis, Santa has had to put his North Pole residence up for auction and is currently looking for a new place in Lapland.

Merry Christmas from the Mr Science Show!

Listen to his podcast here:

And remember to tell us your science highlights from 2008 to go into the running for some sciencey prizes. Answers will also contribute to our year-in-review podcast early in 2009. Let us know here.

The curse of the duck

Cricket fans love their stats. Even the most casual follower can rattle off the batting averages of their favourite players or tell you how many wickets such-and-such a bowler took in the last test. The most passionate followers can recite each scorecard from this year's Wisden.

The recent news of the great Indian batsman Sachin Tendulkar surpassing West Indian Brian Lara's record number of test runs has given maths-loving cricket geeks another opportunity to pull out their calculators and Excel spreadsheets. I'm openly one of these nuts and did just that.

At the time of writing, Tendulkar had scored 12,027 runs across 247 innings, to overtake Lara's 11,953 from 232 innings. After a little investigation, I found that despite his outstanding average of over 54 runs per innings, Tendulkar's most common score in test cricket is ... zero!

This was quite a shock — the most prolific run-scorer in test cricket has been out for nought (a duck in cricket parlance) 14 times, well ahead of his second most common score — which incidentally is the next lowest you can get: one!

This is completely counter-intuitive, so I took this investigation further. Australian cricketer Sir Donald Bradman is universally regarded as the best batsman ever to have played the game. His average, an astounding 99.94, is so far above every other batsman in the history of the game that he is often acclaimed as not only the best cricketer ever, but the best player ever of any sport. His average is so iconic in Australia that the postcode of the ABC (the Australian version of the BBC) is 9994 in every capital city. If it wasn't for the fact that much more test cricket is played nowadays than in the early 1900s, and for World War II interrupting his career for six years, Bradman would have scored many more than the 6996 runs he did score.

So, guess what Bradman's most common score was?

That's right, zero!

Indeed, looking at every innings by the most prolific batsmen in test history from Tendulkar at number 1 to Bradman at number 34, the most common score is zero — and by quite a long way too. The following figures show the distribution of scores from these top batsmen — on the horizontal axis you see the number of runs and the vertical axis measures the frequency of dismissals at a particular number of runs. The first chart shows every score between 0 and 100, and the second uses five-run wide bins to show scores up to 250. The data only include scores where the batsman was dismissed and so does not include not-out scores.

Scores plotted against dismissal frequency.

Scores in bins of five plotted against dismissal frequency.

 

Model cricket

A closer look at these distributions shows that they very closely fit what is known as an exponential distribution. An exponential distribution has the form


In this case y is the probability of being dismissed at score x with λ being constant.
A common trick when looking at distributions involving exponentials is to take logarithms of both sides to get

The graph of this function, plotting ln(y) against x, is now a straight line with slope -λ. If the statistical data fits the exponential distribution, then the plot of the logarithm of the frequency of dismissals against the score at which dismissal happened should look roughly like a straight line.

A straight line fitted to the data. The blue dots represent observed data and the black line represents the model.

A straight line fitted to the data from the second chart above. The blue dots represent observed data and the black line represents the model.

There is a very strong straight line fit in both charts. Using a standard technique called least-squares regression, we can find the straight line that best fits the data. We can determine λ from the coefficient of x in the equation of this line, and in our case this gives λ equal to 0.023.

The mean of an exponential distribution, a sort of average, is 1/λ. In our case this gives a mean of around 43 - the same as we observe in the raw data. One can make the interesting observation that there is no such thing as the "nervous nineties": players do not "choke" and get out in the 90s, nervous before scoring a glorious test century, any more than they get out at any other score. Indeed, you could argue the opposite given the probability troughs at 94, 98 and in the 190s. You can also see that the probability of being dismissed for a duck is higher than you might expect for an exponential distribution.

So what?

Now, so far you might be thinking that all of this is only of passing statistical interest. So what if cricket scores follow an exponential distribution? Well, I'm glad you asked!
Let’s turn for a second to a different distribution, the geometric distribution. You will be familiar with this distribution from a simple 50/50 coin toss. The geometric distribution describes the number of coin tosses you need before a head (or tail) first turns up. The probability of your first head turning up on your kth toss is described as


where p is the probability of a head turning up on each toss, that is, 0.5. The distribution is memory-less, which is one of its key descriptors. No matter what has gone before, even if you have fluked 100 tails in a row, the probability of a head turning up on the 101st throw is still p.

The geometric distribution only works for integer values of k, that is, you can only throw a coin 2, 3, 100 etc times and not 2.5 times. The exponential distribution is the continuous equivalent of this distribution, extending it to work for all numbers, not just integers. Given that cricket batting scores seem to fit a exponential distribution, this means that we can picture cricket batting scores on a geometric distribution with the probability of you being dismissed at score as


Can you spot the profound result here?

Remembering that the geometric distribution is memory-less, you can interpret this as saying that no matter what score you are currently on, you have the same chance p of getting out on that score as you do on any other score! Like a coin toss, the probability of you being dismissed on each score does not depend on what has gone before. A model which assumes that there is no memory is known as a constant hazard model.

This seems to go against every cricketing manual I have ever read. Accepted cricketing wisdom says that a batsman is more dangerous when (s)he "has the eye in" and has scored 10 or 20 runs. Our result seems to suggest that, apart from when a batsman is on 0, you have just as much chance of dismissing him or her on the current score as on any other score.

The next question to ask is, what is the probability of dismissing a batsman on the current score (that is, what is p in the above equation)? The mean of a geometric distribution is


Knowing that the mean of the exponential distribution is 1/λ, and transferring this to the geometric distribution, we get


For λ = 0.023 this gives p = 0.022. Therefore, if you were to turn the television on now and find the cricket coverage, the chance that the batsman you are watching gets out on the current score is 2.2%.

 

Scores near zero

The biggest deviation from the geometric distribution is for scores near zero. According to our data, the chance of being dismissed for a duck is 6.9% — around 3 times more than expected for a geometric (or exponential) distribution. But by the time the batsman has scored two or three runs, the geometric distribution starts to fit well. There is a small peak at four runs, perhaps because you can relatively easily get to four before you become comfortable — it only takes one streaky shot to the boundary. Whilst you can get to three with one shot, you are more likely to have played a few shots and so may be comparatively more "set".

The data and the geometric distribution. The blue dots represent observed data and the black line represents the model.

An analysis of scores near zero has been completed by Brendon J. Brewer from the University of New South Wales in Getting your eye in: A Bayesian analysis of early dismissals in cricket. Brewer indeed found that batsmen are more vulnerable at the beginning of their innings.

By assuming a constant hazard model, Brewer determined the effective average of a batsman before they have scored — that is, assuming a constant hazard model with probability p of dismissal equal to that of their chance of being dismissed for a duck, Brewer determined the mean of this new distribution.

In our data from the best batsmen of all time, dismissal for a duck occurred with a 6.9% chance. The mean of a geometric distribution built around this probability is


This means that even though our batsmen have a mean of about 43, before they've scored they bat like cricketers with a mean of 13.5. Even the best batsmen bat like tail-enders before they get off the mark!

 

Conclusions

What should we take away from this analysis?

The conclusion seems to be that there is a very small window in the beginning of a batsman's innings in which there is a greater chance of dismissal than there ordinarily is. This makes sense — batsmen take some time to acclimatise to the game conditions. But this is a small window — once the batsman has scored about three runs, you have the same chance of dismissal whatever the current score. Interestingly, tiredness does not seem to play a part — the exponential distribution holds well out to 250 runs (quite a few hours of batting).

It should be remembered that this analysis was completed on the top 34 run scorers of all time (5953 innings) and so represents the best ever batsmen. Lesser batsmen are likely to get low scores, so perhaps this window is slightly wider for them. But if we turn to the greatest of the great, Bradman, the window is essentially one run. His effective average before he had scored was a very mediocre nine runs. After he had scored two runs, this effective average had risen to 69. You had to get Bradman out very early!

---

More information

• The data was retrieved from cricinfo during the second test between Australia and India on the 19th of October 2008;
• Not-out scores were removed from the analysis;
• The exponential distribution does break down a little for scores above 250 as there simply isn't enough data;
• Yes, Marc has scored a duck in his cricket career.

Further reading

• Read Brendon J. Brewer's paper Getting your eye in: A Bayesian analysis of early dismissals in cricket;
• Find out more on the maths/cricket blog Pappus' plane — cricket stats.
• Of course, see Plus!

Ep 94: The Geek Pop Virtual Music Festival

Geek Pop is the world’s only sci-pop festival - a free online music event featuring songs about science. The festival brings together science-inspired artists from around the globe in a gleeful celebration of geek culture. In 2009, Geek Pop will take place between 6-15th March.

This week on the podcast I spoke to Hayley Birch, the organiser of Geek Pop, about the festival, where the idea came from and what type of music we can look forward to.

You can subscribe to Geek Pop updates by sending an email to news@geekpop.co.uk with the subject SUBSCRIBE ME RIGHT UP. Or register your attendance at the Facebook event.

We've looked at the various scientific aspects of music in the past on Mr Science, just check out our music label.

Listen to his podcast here:

And remember to tell us your science highlights from 2008 to go into the running for some sciencey prizes. Answers will also contribute to our year-in-review podcast early in 2009. Let us know here.

Ep 93: Communicating Mathematics with the Masses

Some people find mathematics perplexing, whilst others find it beautiful. This week on the podcast, I spoke to two young men professionally employed to communicate mathematics to school-children around the country. Both these guys have a long history in science communication - I got to be good friends with them back in my science circus days.

Jamos McAlester travels Australia with Tenix Questacon Maths Squad, which is an outreach program of Questacon – The National Science and Technology Centre. The Maths Squad aims to inspire students and teachers about maths, and show how science and technology, and in particular maths, play an important role in our everyday lives. The Maths Squad also offers professional development workshops for teachers. Initially started in 1976, it has now visited thousands of towns across Australia, and there aren't too many places in Australia that Jamos hasn't been. The program makes almost 500 puzzle-based activities accessible to students and aims to highlight the broad and narrative nature of maths and its essential and pervasive range of applications. Jamos has a particular love of maths and thinks that people often find maths boring because it is taught out of context:

"calculations are the spelling of maths, not the story."

Marcus Finlay is a proactive, scientifically inclined, primary school teacher from Melbourne. As opposed to most teachers, Marcus inspires his students about science and maths rather than running away from the topics, and lists his class's attempts to build model tsunamis in the classroom as his science highlight of 2008. Back in 2001, Marcus and I wrote The Marco Show about a couple of wizards who sung songs and turned themselves in dogs - you can read about this ridiculous show on Mr Science from 2006.

I spoke to Marcus and Jamos from The Mathematical Association of Victoria's annual conference, both were giving key-note addresses on the communication of mathematics.

Listen to his podcast here:

And remember to let us know your science highlight from 2008. You will go into the running for some sciencey prizes and we'll take a look at your highlights in a podcast episode in early 2009. See the form here.

What are your scientific highlights from 2008?

As we are nearing the end of 2008, and closing in on 100 podcast episodes of the Mr Science Show, I thought it time to ask you the reader/listener about your scientific highlight of 2008. Is it the opening - and closing - of the Large Hadron Collider? Is it the discovery of water ice on Mars? Or perhaps breakthroughs in cancer treatment?

Whatever it is, let us know and everyone that writes in will go into the running for a prize. It'll be a good prize too - either some science magazines or books, something from the Mr Science store, or whatever I can get my hands on. Fill in the form below, leave a comment, or send us an email, and your thoughts will be included in an upcoming podcast reflecting on the science year that was.

THIS COMPETITION IS NOW CLOSED

Going mobile and QR codes

For those of you who need your content on-the-go, we've mobilised Mr Science. You can now see the Mr Science Show in a mobile phone compatible format at m.mrscienceshow.com. And for those of you with an iphone, we have made a site especially for you here. To create these sites, we used MoFuse.

Mobile sites deliberately cut back on data rich content, so pictures are smaller and many of the other widgets, which are easy to download on your desktop but add up on expensive mobile data plans, are disabled. At the moment, as mp3s are disabled, if you wish to download the podcast, you need to do it through the normal desktop version of this site (which you can see on your mobile, but contains all the data-rich stuff). I would like to make it such that if you visit Mr Science on your mobile, your phone immediately redirects you to the mobile site, but alas there is no way yet to do that (well not in blogger, there is for wordpress blogs).


QR Codes

QR Codes (Quick Response Codes) are two-dimensional bar codes (like the one on the left) originally developed by Japanese company Denso-Wave. Having just been to Japan, I can assure you they are very popular there. They are similar to normal bar codes, but are more easily customised and allow you to access internet sites and download content to your mobile without having to type in a URL or go searching through Google.

We created a QR code for the Mr Science mobile site - it's that black and white bar-code looking thing on the left.

To use a QR code, you need a camera in your phone - they all do these days - and you need QR software - grab it from i-nigma if you don't already have it. Open up the i-nigma application and hold your camera near the code. Your mobile then reads and analyses the code and takes you to the page it is encoding. Try it on our QR code - it works on a computer screen.

In Japan, everything from restaurant menus to competitions on train station walls have associated QR codes so you can enter the competition or order from the menu on-the-go. You could use a QR code as a business card if you liked. Various places around the world are starting to use QR codes. The Brooklyn Public Library uses the codes to identity each of their branches, which they then add to flyers and posters. The Powerhouse Museum used QR codes for Sydney Design 08.

For more information, see bibliothekia and ReadWriteWeb.

Climate Change News and closing in on 100 episodes

This week on the podcast, we take a look at the latest climate change news, and start reflecting on the science year that was, as we close in on 100 podcast episodes (this is episode 92).

The news items we discuss this week are:

1. CO₂ build-up in the atmosphere may prevent a coming ice-age. Ice-ages occur roughly every 100,000 years and are possibly due to small shifts in Earth's orbit which change the amount of solar energy hitting the surface. A build-up of CO₂ and its associated heating may warm the Earth so much that the next ice-age is skipped. Humanity has burnt about 300 gigatonnes of carbon from fossil fuels during its existence, and even if only 1000 gigatonnes are eventually burnt (from total reserves of about 4000) then it is likely that the next ice age will be skipped, whilst the next five could be skipped if all recoverable fossil fuels were burnt. For more information, see the story at ABC Science;
2. The bouquet of wine reflects the amount of fossil-fuel derived CO₂ in the air at the time and place of the growing of the grapes. Carbon-14, an isotope of carbon, is made when Nitrogen atoms high in the atmosphere absorb neutrons from space (cosmic rays) . Over time, Carbon-14 decays to Nitrogen-14 , and so fossil fuels, made millions of years ago from decaying organic matter, contain almost no Carbon-14. Therefore, when fossil fuels are burned, the resultant CO₂ is almost Carbon-14 free. As CO₂ is used by plants to grow, the amount of Carbon-14 in the atmosphere at the time of growing is reflected, in this case, in the wine's bouquet. A low level of Carbon-14 means there was a lot of fossil fuel generated CO₂ in the atmosphere at the time of growing. More information on Discovery Science;
3. Wind farms could steer storms. Future mega-wind farms for renewable energy generation could have a massive effect on the weather because the large wind speeds they generate could cause disrupted air-ripples that spread out like waves over massive areas. The waves could even steer storms on the other side of the globe. More information on Discovery Science;
4. Tibetan glaciers are melting faster than ever seen before. The Himalayan glaciers are melting so fast that the usual techniques for dating glaciers can't be used. Glaciers can be dated by looking for traces of leftover radioactivity from US and Soviet atomic bomb tests in the 1950s and 1960s. In the Tibetan samples, there are no signs at all of these tests, and the exposed surface of the glacier dates to 1944. More information at ABC Science;

If you are gazing at the sky this weekend (Thanksgiving to our American friends), then look to the sky to see Jupiter, Mars and the moon all gathered together.

We also start reflecting on the year that was with my much better-half Eugenia, who had to put up with me recording this here little podcast, and then having to listen to the episodes and smile! Her highlights from the year?

• The no-brainer research that said in the UK you should wear thick-clothes, especially denim, to protect yourself from skin-cancer. We never saw the sun over the 2007 summer anyway...
• And on from this show, the Science of Sumo wrestlers - we employed the sumo diet on our recent travels.

Listen to his podcast here:

That time again - Movember

This year, the Mr Science Show has again entered the charity moustache growing contest, Movember. Movember supports men's-health charities, and this year in Australia, all money raised is going towards the Prostate Cancer Foundation of Australia and beyondblue - the national depression initiative. These issues are close to my heart as people close to me have suffered, and even died, because of prostate cancer and depression. Read more about the Fundraising Outcomes.

Did you know:

• Depression affects 1 in 6 men....most don't seek help. Untreated depression is a leading risk factor for suicide.
• Last year in Australia 18,700 men were diagnosed with prostate cancer and more than 2,900 died of prostate cancer - equivalent to the number of women who will die from breast cancer annually.

For more sciencey stuff on moustaches and Movember, here is everything you need:

• The science of moustaches;
• Movember time-lapse from 2006;
• My photos from the 2006 and 2007 events.

To sponsor us, check out our Mo-space or donate directly from this link

Thanks y'all!

Science on Stage

Dr Christopher Pettigrew, a post-doctoral researcher at the Department of Biochemistry in University College Cork, is no stranger to putting science on stage.

With the 2009 AAAS Dance your PhD competition up-and-running, we decided for this week's episode of the podcast to chat to Chris about his experiences in the public performance of science.

Chris has been involved in the communication of many difficult subjects through artistic means, such using interpretive dance to explain the Australian Goods and Services Tax (GST) and DNA . Chris plans on building upon his experiences in Australian theatre whilst in Cork, Ireland. As Dr Pettigrew says:

"Nothing says Double Helix like a rapid twirl."

Listen to his podcast here:

The 2009 AAAS Dance your PhD final contestants have been selected. To read more about them and watch their videos, visit The 2009 AAAS Science Dance Contest homepage.

Do you have any scientific ideas that you would like to see put on stage? Please let us know by leaving a comment here, or by emailing us.

Football Manager Lifetimes

I put this story together for Plus - I'll be writing their new regular Sports Column, which will focus on maths and sport. For more maths stories on sport, see previous Plus stories tagged with sport. And for more on this particular story, see it as published in The Plus Sports Page: Power Trip

Being the manager of a Premier League football club may seem like one of the most glamorous jobs in the world — with the fame comes fortune and the opportunity to travel (well, to Hull, Wigan and Portsmouth anyway). However, as far as job security goes, football managers live on the edge. Their terms can be terminated almost on a whim by their club's owner, and they live and die by their team's results.

It would seem that there is no way to predict how long their tenures will be. However, a collection of researchers from the UK, Singapore and the US have found that there may be a strong mathematical trend underlying how long football managers stay in their jobs.

Toke S. Aidt, Bernard Leong, William C. Saslaw and Daniel Sgroi found that the distribution of tenure lengths for managers of sporting teams in many countries obey power laws. Power laws are fascinating because they arise in a surprisingly large number of naturally occurring phenomena, such as the size of cities, stock market returns, cook book ingredients and even how many times certain words are used in long books.

A power law has the form

where x and y are variables and a and b are constants. The exponent b is usually negative, so y decreases as x increases. In the case of football managers, the researchers found that

where t is the length of a football manager's career and n is the number of managers dismissed at that time of their career. The actual values of the constants a and b vary from country to country and league to league, with the exponent b lying somewhere between -2 and -3 in all cases.

To derive this formula, the authors plotted tenure lengths of real managers against their time of dismissal and then set out to find the curve that best describes the data. In fact, to make things easier, they looked at logarithms, which turn a curve of this form into the straight line

For tenures greater than one year, they found that in the English Premier League, football leagues across Europe, and American football and baseball competitions, there is a straight line of this form that fits the data. Moreover, the fit is statistically significant, that is, it’s not just due to chance.

The following graph is for English Premier League managers between 1874 and 2005.

The logarithm of the length of managers' career plotted against the logarithm of the number of managers dismissed at that time of their career. The data can be approximated by a straight line and the fit is statistically significant. The data come from the English Premier League between 1874 and 2005.

But what does all this mean?

As we mentioned earlier, power laws are compelling as they can emerge from simple mathematical rules — the power law is often a macroscopic outcome of microscopic interactions between the players in the system (in this case football managers, the team, club owners and fans, etc). In fact, power laws are often seen as the signature of complexity. In the 1980s scientists found that there are dynamical systems based on simple rules which, through self-organisation, bring themselves into extremely sensitive states, where even the smallest change can cause wide-ranging and unpredictable chain reactions.

An often quoted example of this phenomenon involves a pile of sand. When you sprinkle sand on a table, a pile will build up and after a while reach a maximal slope: any additional grain of sand will cause avalanches whose number and size are impossible to predict. Such a sensitive state is called a critical state and this behaviour is called self-organised criticality. It is an interesting phenomenon, because it may explain "spontaneous" emergence of complexity in nature, which is not a result of someone forcing the system.

When a system has reached a critical state through self-organisation, it can often be described by power laws. In our sand example, the size distribution of the avalanches follows a power law. Power laws reflect complexity because they are similar on all scales. Suppose that the number n of avalanches of size s is described by the power law

for some constants a and b. Now multiply s by a large number c, so you're now looking at large avalanches. These then follow the power law

which, apart from the constants involved, is essentially the same as that for smaller avalanches - the same type of behaviour occurs on all scales.

Given that the power law highlights the fact that there is something interesting going on, the researchers set out to find out what it was. What are the simple rules of football management that govern this system, and is there self-organised criticality?

The model

The authors constructed a model which includes a manager's reputation — this is either enhanced or diminished, depending on the result of each match. The core of the model is a round-robin tournament with 20 teams playing each other once at home and once away — just like in the Premier League. The probabilities of win, lose and draw were modelled as 37%, 26%, 37%, respectively — these probabilities are those observed in the English football league between the years 1881 and 1991 and are assumed to be independent of the managers involved.

The model starts with 20 randomly selected managers, each with a given reputation and tenure. (With a nod to realism, we will henceforth assume that all managers are male.) The initial reputation of each manager is described by a positive whole number, which is chosen at random from the numbers between the firing threshold and the poaching threshold (more on these in a moment). Each manager also starts with a random tenure length between 1 and 40 years. The managers gain reputation (+ 2 points in the model) every time their teams win, and lose reputation (-2 points) when their teams lose. There are no points for draws. Each game has equal importance and so each result is equally important for a manager's reputation.

The length of a manager's tenure depends on how his reputation evolves. Termination of tenure can occur for four reasons:

• The manager loses his job when his reputation falls below the firing threshold — that is, he is sacked;
• The manager is poached by another club when his reputation reaches the poaching threshold — that is, he gets a better deal;
• The manager retires if he gets too old (another parameter that can be varied);
• The manager's team is relegated to a lower league because it has the lowest reputation at the end of the season — the team is demoted out of the league.

When a manager leaves the system — that is, he is either fired or poached, relegated or retired — his place in the league is taken by another manager with tenure length of zero and a random starting reputation.

With these rules in place, the researchers ran many simulations, varying the random parameters in each run. Such a process is known as a Monte-Carlo simulation. They recorded the distribution of tenure lengths corresponding to one hundred years of competition. They found that for a very broad range of starting parameters, the model produced a tenure length distribution statistically indistinguishable from a power-law distribution. Similar results were obtained for different probability distributions of win, loss or draw. However, the researchers also found that power laws only emerge when a win enhances reputation by the same amount as it is decreased by a loss, and when each match has equal importance. The latter makes sense if you think that the aim of a Premier League team is to maximise its profit: you need to fill the stadiums and make as much advertising revenue as you can at each game. And as the Premier League is a first-past-the-post competition, each win has equal worth on the league table, with position on the table more than anything guaranteeing further advertising and merchandising returns.

Coming back to self-organised criticality, the researchers admit that their model does not prove the existence of this phenomenon in the world of sport. In fact, the model is not quite as self-organising as it could be, since certain parameters need to be artificially fixed at the outset. They do believe, however, that certain other factors point in the direction of self-organised criticality. The Premier League, they postulate, follows the Red Queen principle: it is an arms race where constant development is needed simply to compete. This explains why once a league has reached a self-organised critical state, it might stay there for a prolonged period of time. It is simply too difficult for a team to shake up the system, given that they are already in a process of continual change in order to stay with the pack. The term Red Queen comes from Lewis Carroll's Through the Looking-Glass in which the Red Queen says: "It takes all the running you can do, to keep in the same place".

What the results surprisingly show is that ability and talent, although obviously playing some role, do not play a major role in a manager's success. His survival is far more determined by the sacking and poaching thresholds and simple randomness in his team's results. 2007 Chelsea manager Avram Grant is a good example of this: as he started his tenure with a low reputation, despite his team's good results, probabilities took their toll and he was sacked at the end of the season.

In any case, it's hard to feel sorry for prematurely sacked Premier League managers when their average salaries are over £2 million.

For more info, see the following:

• The paper A power-law distribution for tenure lengths of sports managers has appeared in the journal Physica A;
• The Plus article Understanding uncertainty: The Premier League finds more randomness in the Premiere League;
• There is more about sand pile models in the Plus article Like sand through the hour glass;
• And about power laws in the Plus articles The mystery of Zipf, Network news, and Beating bird flu with bills.

Here's why we need science communicators

From the Border Mail, Letters 01/10/2008

When I was a kid, we never had drought after drought.

Then we started with daylight saving. We started with a little bit, but now we have six months of the year daylight saving. It has just become too much for the environment to cope with.

It is so logical, for six months of the year we have an extra hour each day of that hot afternoon sun.

I read somewhere that scientific studies had shown there is a lot less moisture in the atmosphere which means we get less rain.

I believe this one hour extra sun is slowly evaporating all the moisture out of everything. Why can't the government get the CSIRO to do studies on this, of better still, get rid of daylight savings.

They have to do something before it is too late.

CHRIS HILL, Albury

... And that's why we need science communicators!
This has also been reported in Failblog and somewhere in the smh.

Halloween Science Special

As it's Halloween, here is the Halloween news that I presented on Diffusion Science Radio this week. Diffusion can be heard on Monday nights at 6.30pm on 2SER 107.3 in Sydney, at various times across Australia on stations affiliated with the Community Radio Network, and on the Diffusion podcast.

1) Halloween, Candy and Science

What's worse: eating all the lollies collected on Halloween night at once, or spreading this out over the coming days and months?

When it comes to your teeth at least, it is far worse to ration your lollies all through the day, day after day than it is to gorge it all at once. Mark Helpin, a pediatric dentist at Temple University, says that snacking on candy keeps your teeth bathed in enamel-corroding acid, which is produced by bacteria feeding on the sugar in your mouth.

When you cover your teeth with sugar, oral bacteria cause a rise in acidity levels. This is neutralised when you brush your teeth. Even if you don't brush, saliva will eventually wash away the sugar and starve the bacteria. If you continually snake on chocolate and other lollies, the level of acidity stays constantly high, and this can lead to tooth decay.

Helpin also thinks that potato chips are just as bad, or worse, than lollies. Acid-producing bacteria feed on carbohydrates in potatoes, which are far more sticky than lollies and so hang around longer on your teeth. This poses an even greater risk for tooth-decay.

More information on ABC

2) Bigfoot revealed to be Halloween Costume

There's been a recent downturn in the fortunes of those hunting for Bigfoot, which a supposed frozen corpse of the animal turning out to be a Halloween costume.

SearchingforBigfoot.com owner Tom Biscardi had paid an estimate $50,000 to Matthew Whitton and Rick Dyer for their frozen Bigfoot "corpse". Biscardi also hired Sasquatch detective Steve Kulls to check out the specimen.

Kulls was not a happy man, and neither, it turns out was Biscardi, especially after Whitton and Dyer ran off with his money.

"I extracted some [hair] from the alleged corpse and examined it and had some concerns," Kulls writes. "We burned said sample and said hair sample melted into a ball uncharacteristic of hair. Within one hour we were able to see the partially exposed head. I was able to feel that it seemed mostly firm, but unusually hollow in one small section. This was yet another ominous sign."

"Within the next hour of thaw, a break appeared up near the feet area. ... I observed the foot which looked unnatural, reached in and confirmed it was a rubber foot."

When Biscardi found out, he called Whitton and Dyer at their California hotel, who confirmed the hoax. However, when Biscardi went to look for them, they had disappeared with his money, and plenty of his dignity.

More information on FoxNews.com

3) Vampire Moth

A population of vampire moths has been found in Siberia that entomologists suggest may have evolved from a purely fruit-eating species as there are only slight differences in their wing patterns from the herbivorous cousins, Calyptra thalictri.

When the Russian moths were experimentally offered human hands , the insects drilled their hook-and-barb-lined tongues under the skin and sucked blood.

Entomologist Jennifer Zaspel from the University of Florida said the discovery could shed light on how indeed caught a fruit-eating moth evolving blood-feeding behavior, it could provide clues as to how some moths develop a taste for blood.

It may be that blood-feeding in insects evolved from feeding on tears, dung, and pus-filled wounds.

"We see a progression from nectar feeding and licking or lapping at fruit juices to different kinds of piercing behaviours of fruits and then finally culminating in this skin piercing and blood-feeding," she said.

In addition, only male moths exhibit blood-feeding, which means that maybe its so the males can pass on salt to females during sex. This could provide a nutritional boost to young larvae that have sodium-poor diets.

More information on National Geographic

4) Trick-or-treat safety tips

Children are twice as likely to be hit by a car and killed on Halloween than on any other day of the year - so take care! More information at AJC

Listen to his podcast here:

We're now on Twitter

For those of you who use Twitter, we are now there and embracing it, so come along and become a follower.

We've blogged quite a bit about Web 2.0 applications (most recently on data-mining and Last.fm). Twitter is a micro-blogging service that allows its users to send and read other users' updates (otherwise known as tweets), which are posts up to 140 characters. When you 'follow' someone, you can see their updates, and it has become quite popular. Even politicians are getting on board, with Australian Opposition Leader Malcolm Turnbull opening an account (whether he has time to continually update his Twitter status, or it's one of his staffers, is another question).

Twitter has become a surprisingly powerful platform, breaking the news of the Sichuan earthquake in China with SMS messages well before the conventional media arrived. This was also the case in the Virginia Tech shootings.

So if you would like to follow me on Twitter, I am @westius. You can use this site's email address to find me. See you there.

So, you think you can dance...

We've explored the overlap between music and science before, but what's dance got to do with science?

The 2009 AAAS Science Dance Contest is just around the corner, so if you're a scientist with a deep longing to express your innermost scientific thoughts through dance, then this is for you. The contest is open to anyone who has (or is pursuing) a PhD in any scientific field.

What you need to do is:

1. Make a video of your own PhD dance;
2. Post the video on YouTube;
3. Email your name, the title of your thesis, and the video link to gonzo@aaas.org by 16 November 2008.

On 17 November 2008, a total of four winners will be chosen from the following categories:

1. Graduate Student
2. Postdoc
3. Professor
4. Popular Choice

If you are lucky enough to win one of these categories, you will need to provide a single peer-reviewed research article on which you are a co-author. You will be paired with a professional choreographer and over the following weeks you will help your choreographer understand the article (via e-mail and telephone). Then the four choreographers will collaborate to create a single four-part dance based on the winning research articles.

You will then be an honoured guest at the AAAS Annual Meeting in Chicago, Illinois, where on 13 February 2009, you will have front-row seats to the world debut of "THIS IS SCIENCE" - your dance creation. Accommodation in Chicago will be provided, and grants are available for travel expenses.

To read more about last year's competition, see sciencemag and gonzolabs.

Stay tuned to this website as we are going to follow this contest, and I have already roped in a couple of my PhD friends to enter - or at least, they're thinking about it...

Japan and the Sumo Diet

Travelling Japan and Korea was a wonderful experience - magnificent sights, great food, interesting people and of course, of interest to this blog, science.

I have finally put together the Japan / Korea podcast, which you can listen to here. It was recorded whilst we travelled from Tokyo to Fukuoka on the shinkansen (bullet train), Fukuoka to Busan in Korea on the JR Beetle (ferry), and Busan to Seoul on the KTX (high-speed train) - all fantastic methods of transport that put Aussie transport links (and for that matter UK ones) to shame.

The first topic we tackled was the Sumo Diet - how do Sumo Wrestlers get so big? And why? We were lucky to catch the September Grand Sumo Tournament in Tokyo and I was astounded by how big and strong these guys are. What's their secret to rapid weight gain? Well, here's what you need to do:

1. Skip breakfast. Often people who try to loose weight skip breakfast, but it's actually the worst thing you can. After 8 hours of sleep, your body craves food. By depriving your body of food, you keep your metabolic rate low;
2. Exercise on an empty stomach. If you exercise with no food to burn off, your metabolic rate lowers even more in order to conserve the energy you have. This helps increase your muscle but not burn off too many calories. This point is rather open to debate as to whether it works;
   
3. Sleep after eating. After a massive lunch or dinner, have a sleep. This means you wont burn off all those calories you just ate. This is a major factor;
   
4. Eat big in the afternoon and evening. Going to bed with a full stomach makes your body release a rush of insulin, storing some of your previous intake as fat instead of in muscles and organs;
5. Eat in a social atmosphere. An unreferenced report says that when you eat with others, you eat more than 44% more than when alone. I'd believe that.
   

You can find more info on this diet here.

At the heart of the diet is a hearty stew called chankonabe. It is a communal one-pot simmering stock-based casserole, into which you dip chicken, pork, mushrooms, carrots, potatoes, radish, lotus root and onions as if it were a massive meaty fondue (which I guess it is). You can drink the left over stock. Now this doesn't actually sound unhealthy - indeed it actually sounds pretty nutricious, but if you eat a lot of it, and then have a sleep, you can get very big. And they do eat a lot. A wrestler named Takamisugi was revered for eating 65 bowls in a single sitting.

We tackled a number of other topics in the podcast, so tune in. And feel free to leave any comments you like - I would love to hear from you, especially if you have tried the Sumo Diet...

Listen to his podcast here:

Last.fm, data mining and mashups

I've recently been putting together a Guide to Web 2.0 for The Helix Magazine and one of the most interesting aspects has been exploring the various mashups and applications of Last.fm.

Last.fm is a brilliant online music service and currently my favourite "web 2.0" application. By downloading a plugin for itunes (or whatever music player you have) that "scrobbles" each song you play (that is, tells Last.fm what you are listening to), a picture of your music taste builds up, and people with similar listening tastes are found. Artists are recommended to you according to your tastes, charts of your songs built up and "radio stations" perfectly tailored to you can be streamed online. But it is better than radio as there are no ads and you like every song.

By the way, I am westius on Last.fm.

Millions of songs are scrobbled every day by Last.fm users. This data helps Last.fm develop a massive database of user music preferences, and because of it's API, it is possible to access Last.fm information and develop interesting tools.

As users can tag their music with genres that they think aptly describe their songs and artists, it is possible to determine your own tag cloud of musical preferences. Using an excellent script at anthony.liekens.net, I came up with my own tag cloud, as you can see here.

It is possible from such tag clouds to examine how listeners fall into different categories through a process known as Data Mining. Data mining is essentially the process of sorting through enormous amounts of data and picking out the relevant stuff. Using principal components analysis - a mathematical technique which reduces multidimensional data sets to lower dimensions for analysis - and k-means clustering - an algorithm to cluster n objects into k groups - Liekens came up with 5 broad groups of Last.fm listeners:

1. Electronic/pop
2. Rock
3. Indie
4. Metal
5. Hip-hop
   

Clearly this list does not reflect everyone on Last.fm (where are the classical music listeners?), but it does reflect the majority. I was surprised that Indie is a group in itself and am intrigued by the bundling of electronic and pop together - there are some tweaks to the maths you can make that could come up with different groups, and better results might be possible with a bigger data set . Hip-Hop listeners were the most clearly defined group. You can read more about the maths and how these groups are separated in the original article.

Another interesting thing you can do is compare your music tastes to your friends. This pic is a difference cloud comparing my music tastes with that of my good friend intranation. We have a roughly 40% similarity in music genre tastes, with the green tags those that I have more of in my collection, and the red those genres that intranation listens to more than me. No real surprises there.

Mashups are all the rage at the moment. The term refers to web applications that combine data from more than one source into a single integrated tool. For instance, domain, an Australian real-estate site, adds data from Google Maps to provide location information. My current favourite Last.fm mashup is idiomap. idiomap is a digital music magazine that personalises its content according to your interests in music, which it learns from your Last.fm profile. It gives you stories and reviews of the artists and genres you like, helps you discover new music and mashes in video and audio from youtube and other sources. idiomag aggregates music articles from over 100 different sources. You can also tweak the articles you like so if you receive something you don't like, you won't get it again. I subscribe to the RSS feed of my personalised idiomap magazine and so far its been great and has included reviews of music DVDs of artists I like and schedules of when bands will be playing and appearing on TV. Good stuff.

I will probably put out a few more blogs like this as I explore this world of mashups. And for podcast listeners, yes hopefully I will get one of them out soon too!

Why Hollywood stars should stick to acting

It's funny being on holiday, removing yourself from everything and deliberately avoiding the news, and then returning home to find things largely the same as they were. Sorry for the lack of posts and podcasts, I have been travelling back to Australia through Japan and Korea - stay tuned for a podcast from the trip.

One of the things that has not changed is our own Russell Crowe. He may have played a maths genius in A Beautiful Mind, but his maths skills don't say much for his Aussie education.

We all have our theories on what is causing the current financial meltdown - my theory is it's the Australian cricket team (see this story for why).

But Crowe has an interesting solution - give America's entire population of 300 million $US1 million each.

His thinking was that a $300 million outlay would only be a fraction of the $US700 billion bailout package that President Bush proposed (read that carefully and you will spot the mistake).

Crowe told Jay Leno, "So, here's the thing: They're looking for $700 billion, right? Which is a good chunk of change... But I was thinking if they wanna stimulate the economy, get people spending, let people look after their ... mortgage. I think you take the first 300 million Americans, if that's the population at this point in time, give everyone a million bucks.''

The problem is that the Crowe Plan actually only gives $1 to each American, not $1 million, and if the Crowe Plan to instantly make each American a millionaire went ahead, it would cost $300 trillion - more than the US annual gross domestic product. The Iraq War only cost $3 trillion.

Crowe clearly didn't do his method acting for A Beautiful Mind.

Edit:
Funnily enough, he could be correct if he was using the Long scale where one billion is actually one million million (not one thousand million). Most of the English speaking world uses the short scale, but much of the world uses the long scale - so perhaps we can forgive him. Even NASA has mucked up unit conversions, loosing a Mars orbiter in 1999.

Reflections on the London Science Blogging Conference

With the London Science Blogging Conference finished up, it was time to reflect on the good, the bad and the ugly of science blogging.

I chatted to Lisa Bailey, who blogs about science at Bridge 8, about what we learnt at the conference, how blogging can be used effectively to communicate science, the challenges laid down at the conference - including the challenge to get senior scientists to blog - and where science blogging might be going from here.

For more from the conference, check out the Mr Science podcast from our evening in London science pubs.

This is my last post and podcast for a couple of weeks as I travel home to Sydney after an absolutely wonderful 18 months in London. I will miss all the new friends I have made, and all the old friends I have met again. London, it's been emotional. See y'all in Sydney and if any one wants to sponsor a visa for me to come back, just get in contact!

Listen in to my conversation with Lisa here:

Bacon vs. Erdos - it's a numbers game

Remember movie star Kevin Bacon, who fought so bravely for our right to dance in Footloose?

His dance activism aside, Bacon is probably best known for spawning the trivia game Six degrees of Kevin Bacon. The game is based upon the assumption that all actors can be linked to Bacon through their film roles in six steps. For example, Brad Pitt starred with Bacon in Sleepers, so he is connected by one film and has what is known as a "Bacon number" of one. In Top Gun, Val Kilmer starred with Tom Cruise, who starred with Bacon in A Few Good Men, so he has a Bacon number of two.

This theory that every actor can be connected to Bacon within six steps concerns the mathematical field of small world phenomena. Stanley Milgram first suggested that everyone on Earth is connected by a surprisingly small number of people when working at Harvard University in 1967. He sent packages to 160 random people in Omaha, US and asked them to forward the package to a friend or acquaintance who they thought would bring the package closer to a set final individual, a Boston stockbroker. The letter stated, “If you do not know the target person on a personal basis, do not try to contact him directly. Instead, mail this folder to a personal acquaintance who is more likely than you to know the target person.”

Milgram’s theory was that everyone is connected by on average six people – that is, even the Prime Minister is connected to rickshaw drivers in Thailand by six people. Whilst this may seem astounding, think about how many people you come across during your lifetime. Whilst I have never met the Prime Minister, I have met my local member of parliament which means I am only two steps from Rudd. Imagine that my local member has a son who travelled the world and visited Thailand – now the rickshaw driver is connected to the PM in only three steps. You only need to meet one well-connected person to be connected to almost everyone!

The concept of the Bacon number sprung from a similar idea surrounding the mathematician Paul Erdos. Erdos was an immensely prolific Hungarian mathematician who worked on a variety of problems from number theory to probability. He collaborated with so many other scientists that mathematicians invented the "Erdos number". You have an Erdos number of one if you directly collaborated with Erdos on a paper, an Erdos number of two if you worked on a paper with someone who collaborated with Erdos, etc.

The amusing consequence of all this is the "Erdos–Bacon number" which is the sum of your Erdos number and Bacon number. You would think that these two numbers would be completely separate – not many Hollywood stars have authored mathematical papers. However, thanks to a few movie and TV cameos, astronomer Carl Sagan has an Erdős–Bacon number of nine, whilst theoretical physicist Stephen Hawking’s is seven. And most interestingly, thanks to her authorship of psychology papers during her Harvard degree, actress Natalie Portman also has an Erdos–Bacon number of seven!

Thanks to the fact that I was in our high-school production of Grease with my brother James, who had a role as an extra in the Australian gritty crime show Wildside, which starred Rachael Blake, who was in Derailed with Jennifer Aniston, who was in Picture Perfect with Bacon, I have a Bacon number of 4.

There are probably a few ways I could track back my Erdos number, but the easiest way is through Plus which I co-edit with Marianne Freiberger, who has an Erdos number of 4, making mine 5. There is probably another route back to Erdos through my chemistry work, but that is hard to track.

Therefore, my Erdos-Bacon number is 9. See if yours is lower. Here are some links to help:

• Calculate your Bacon number at the Oracle of Bacon;
• Calculate your Erdos number the American Mathematical Society.
  

Crawl of London Science Pubs

Last night I attended the most interestingly themed pub-crawl I have ever been on. It was a science crawl of London pubs.

The pub crawl was held in association with the inaugural science blogging conference, Science Blogging 2008: London, hosted by Nature Network, in collaboration with the Royal Institution. The aim of the conference is to bring together science bloggers from around the world to discuss the pressing issues in science, science communication, publishing and education. What role does blogging play? I am going tomorrow, so come and say hi!

We went to four different pubs, starting with the Jeremy Bentham, then moving to the Museum Tavern, the Ben Crouch Tavern and the John Snow.

Our tour guide for the evening was Londonphile and editor of the Nature Network, Matt Brown. On this week's podcast, I chatted to Matt about the four pubs we attended. A few other science bloggers also pop up in the podcast, and they are:

• The Questionable Authority,
• Frank Norman,
• Deep thoughts and sillyness,
• Grrl Scientist.
  

Listen here:

Your favourite fictional scientists - the podcast

Emmett Brown vs Dr Who
Professor Honeydew vs Beaker
Q vs Dr. Evil

And Mr Science vs The Ordinary Guy?

This week the podcast has joined forces with the Brains Matter podcast to discuss the topic of our favourite fictional scientists. We looked at the poll and at your suggestions, then chatted at length over Skype about what turned out to be quite an interesting topic. There is certainly scope for more of these joint podcasts - especially if I get my recording gear together, unfortunately my computer overloaded a bit so some of my bits are a bit clipped.

I will leave the poll open (see right) for a while so please continue to vote (if you can't see it, follow this link). Also see our original story on fictional scientists, and if we have left someone off that you like, please leave a comment.

This podcast also exists in a slightly different form on the Brains Matter website, and you can never have too many Australian science podcasts (although The Ordinary Guy is from Melbourne, but we wont hold that against him...), so check it out.

Listen here:

Modelling Olympic Gold

After every Olympics, there is speculation about which country performed best. Should we really be surprised when China, with its huge population, and the US, with its combination of high GDP and population, top the medal table? Can we take a look at the medal tables and see which countries did indeed perform better than expected?

This is a shorter version of an article I wrote up over at Plus, so to read more, especially about some of the maths involved, see the article Harder, better, faster, stronger.

In terms of total medals won, the same five countries topped both the 2000 Sydney Olympics table and the 2004 Athens Olympics table:

Position	Country	2000 Medal Count		Country	2004 Medal Count
1	United States	92		United States	103
2	Russia	88		Russia	92
3	China	59		China	63
4	Australia	58		Australia	49
5	Germany	56		Germany	49

By-and-large the same countries rise to the top each Olympics, but a quick look at the medal tables seems to suggest two obvious variables that may play a part in a country's Olympic success — population and Gross Domestic Product (GDP). A high population gives a country more athletes to draw from, while GDP could be assumed to represent a country's prosperity, with a prosperous country more likely to spend money on frivolous activities such as sport. Adjusting for population, we see that the top 5 countries have changed, except for Australia, who has over-performed for its population:

Position	Country	2000 Medal Count	Population ('000s) per medal		Country	2004 Medal Count	Population ('000s) per medal
1	Bahamas	2	142		Bahamas	2	150
2	Barbados	1	259		Australia	49	406
3	Iceland	1	273		Cuba	27	419
4	Australia	58	324		Estonia	3	447
5	Jamaica	7	379		Slovenia	4	503

India, with its huge population, under-performed in 2004, with one medal per one billion people, however we may expect with its rising GDP that it could come near the top of future lists. Looking at GDP, we find a new top 5, with Australia dropping out, but Cuba, Jamaica and the Bahamas again performing well:

Position	Country	2000 Medal Count	GDP ($ '000,000s) per medal		Country	2004 Medal Count	GDP ($ '000,000s) per medal
1	Cuba	29	597		Cuba	27	1170
2	Jamaica	7	1257		Jamaica	5	2042
3	Georgia	6	1867		Bahamas	2	2550
4	Macedonia	1	2100		Georgia	4	3045
5	Bulgaria	13	2585		Eritrea	1	3300

Looking at simple plots of medal tally against population and GDP for the 2004 games, it can quickly be seen that linear models of these variables will be unsatisfactory:

The extreme values of GDP and population suggest that logarithms should be used. This makes practical sense — a country with a high population does not get to enter more athletes in the Olympics than lowly populated countries, and whilst a high population gives a strong base from which to draw quality athletes, as population increases, this effect will diminish. With regards to GDP, countries occasionally produce athletes with so much natural ability that no amount of money spent on training the opposition could defeat them. Findings in the report Do elite sports systems mean more Olympic medals? by Simon Geoffrey, Martina Kerim, Peren Arinb, Nitha Palakshappac and Sylvie Chettyd from the Department of Commerce at Massey University back this up, with the authors suggesting that "the extraordinary talent required in winning a gold medal cannot be surpassed by the employment of an elite sports system."

Looking at the countries that received more than 15 medals in 2004, plots of the logarithm of medal count against the logarithms of population and GDP show a linear relationship. Using linear regression — a form of analysis that fits a straight line to the data by minimising the distances between the data points and the fitted curve — we can find a straight line that fits well. We found that the R² values of this fit (R² is a statistical measure of correlation between 0 and 1) are above 0.5, suggesting that, while not quite high enough to prove a correlation, we may be on to something:

Using a linear combination of the logarithms of GDP and population, we can come up with a fitted line:

We can see that Cuba, Australia and Russia all fall above the line of best fit and so compared to the other countries who received more than 15 medals, achieved well. This could be explained by Cuba's famous tradition of boxers and the spending of Australia and Russia on sport.

The danger with any such fitted model is that you can fit anything to anything after the event — the challenge is to come up with a worthwhile representative model that can not only let teams know how they are doing now, but can predict how they may do in the future.

In the paper Who wins the Olympic games: Economic development and medal totals, Andrew B. Bernard and Meghan R. Busse from The National Bureau of Economic Research developed a model that includes population, GDP, whether the country was the Olympic host and whether the country was formerly part of the Soviet Union or eastern block. They found that countries win 1.8% more medals when host than otherwise, and similarly, found that former Soviet Union or eastern block countries, because of their forced mobilisation of resources towards sport, and countries with planned economies, won more than 3% more medals than equivalent western countries. Their model is formulated as:

where M is a country's medal count, N is the population, Y is the GDP, C, alpha and beta are constants, and Host, Soviet and Planned are constants equal to zero or some value depending on whether the country was the host, part of the Soviet block, or had a planned economy.

In their more developed models, the authors included terms to represent how countries performed at previous Olympic games — perhaps to represent the experience gained by athletes competing at multiple games. Their overall conclusion is that whilst GDP is the best single variable for predicting medal tallies, other factors such as being the host country need to be included. Indeed, their model predicted that Australia would win 17 more medals than otherwise when it hosted the Sydney Olympics — the model was only one short of the actual 18 extra medals Australia did win.

With this in mind, it is hard to look past China, as host country and with vast amounts of money pouring into Olympic sports for just this occasion, topping the medal tally.

Data from the World Bank and the International Olympic Committee was used in the analysis. Due to doping scandals, the medal tables may change but are accurate at the time of writing.