pension333: Olympics 2012: the alternative medals table

We all know who's in the lead on the medal tables - but what would happen if you looked at them by population size, or GDP - or even compared to the number of athletes in each team? A team of statisticians has worked with us to bring you a new way of judging the Olympic Games
? Click here for the alternative medal ranking table
? Live: Medals adjusted for population
? Live: Medals adjusted for GDP
? Live: Medals adjusted by team size

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How do you measure a team's performance in the Olympics? The traditional way is to just count up the number of medals won. And the result? The biggest countries always come top: the Olympic 'superpowers' of the US, China, Russia, UK, Australia and Germany.

But what if the totals took account of factors that must have an influence, such as the size of a country's population or its economic power, or compared it to the size of the athletic team in London?

The Royal Statistical Society and the Datablog have teamed up with four statisticians at Imperial College, London, to help us work out how those key factors might change the league table. By 'weighting' the medals, what happens to the results?

The team, Christoforos Anagnostopoulos, Giovanni Montana, Axel Gandy and Daniel Mortlock, looked at previous olympics and traditional indicators such as the output of a country's economy (GDP), the size of its population - and also ways to weight the score by the size of each country's Olympic team.

What's the rationale? Take the 2008 results. The Bahamas had a population of approximately 334,000 in 2008, whereas the USA had 304,000,000 - almost 1,000 times larger. And yet the Bahamas won two medals, whereas the US 110 ? 55 times as many. "Taking population into account," says Anagnostopoulos, "It no longer seems obvious that the US should rank higher than the Bahamas. The intuition is that the US had a larger pool of possible athletes to choose from, and consequently it makes sense that it should do better, too".

So, what will the results look like? Says Anagnostopoulos:

The simplest approach is to divide the number of medals by the population of each country. We will however look at other types of indices that might be harder to interpret directly. Consequently, to make the league table interpretable without reference to the underlying index, we express the results as a (weighted) medal count. As the Games progress, for each medal type (Bronze/Silver/Gold), we redistribute the medals that have been already won, taking into account the country's population: for a small country, one medal will be worth more than for a larger country, and it may therefore end up with 2 or 3 medals, whereas the larger country "loses" some of its medals in order to correct for the advantage afforded to it by way of its larger population. The resulting medal count will depend on the relative sizes of the countries of the medal winners, and may change as more medals are added onto the database. We do the same for silvers and golds, as well as for total medal count

GDP, is another obvious one to re-size on, particularly when you consider how expensive sport equipment and training is. Moreover, since GDP also grows with population size, it implicitly also takes into account population size.

Although penalising larger wealthier countries may seem intuitively "fair", our statistical team invites us to think harder about the potential arbitrariness of penalties and how they can be selected objectively. Anagnostopoulos explains:

We have been thinking of GDP (or population) as an "advantage" that needs to be "corrected for" by penalising. This however involves an arbitrary decision of how much to penalise by. A statistician would take a different, more objective view, where GDP is a factor that can, to some extent, explain the performance of various countries. A different, more objective view, would interpret GDP as a factor that can, to some extent, explain the performance of various countries. Once this explanatory potential is exhausted, what 'is left' (the statistical jargon for this is 'residual') can be interpreted as 'GDP-corrected' athletic skill - a purer measure. Crucially, we may then rely on sound principles of statistical modelling to determine fairly conclusively which index is the one that maximises the explanatory power of GDP (and/or population) in this context. The resulting measure is no longer a simple ratio, but a variant of a log scale, which carefully balances the numbers in a fairly complicated way. When the games are over, we will be able to analyse the results based on this work

Reassuringly, however, the main qualitative conclusion of the earlier league tables seems to persist: for instance, in 2008, Cuba comes top, with 24 medals but a small population and GDP. Not all "superpowers" are banished, but some are: Australia, China, and Russia maintain their positions in the top 20, whereas the UK and the US are no longer featured.

Team size is also a factor - and one which our team says may be a better indicator than either GDP or population as it has already taken that into account by team selection.

And this is what those results look like for for population and GDP, compared to team size: "A sharper linear relationship is evident after taking logarithms. Further analysis can fine-tune the index to capture as much as possible the effect of GDP and/or population on performance." says Anagnostopoulos.

The full data is below - and we will update it throughout the Games. What can you do with it?