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**Is there a correlation between Premier League wages and league position?**

The Guardian has just released its 2012-13 Premier League season data analysis – which shows exactly how much each club in the Premier League spent on wages last year (see the bar chart above). This can be easily plotted on a scatter graph to test how strong the correlation is between spending and league position. (y axis is league position, x axis is wage bill in millions of pounds).

The mean spending on wages is 89 million pounds. Our regression line is y = -0.08x + 17.52. We can see some of the big outliers are QPR (with a big wage bill but low premier league position) and Everton (with a low wage bill relative to others who finished in a similar position).

The Pearson’s product moment correlation coefficient (r) is -0.73. This is negative because in our case league position is numerically lower the higher up the league you are. This shows a pretty strong correlation between league spending and league position. An r value of -1 would be a perfect correlation in our case, whereas 0 would be no correlation.

**Is there a correlation between turnover and league position?**

We can also see what the correlation is between league position and overall club turnover (see the bar chart above). Here we can see there is a huge gulf between the top few clubs and everyone else in the league. There’s only 40 million pounds difference between the bottom ranked club for revenue Wigan and Newcastle, with the 7th biggest revenue. But then a massive jump up to those with the top 6 revenues.

This time we have a mean turnover of 128 million pounds and a regression line of y = -0.05x + 16.89. The Pearson’s r value this time is r = -0.79, so there is a slightly stronger correlation than from wages – and this is a strong correlation overall. So, both wage bills and turnover provide a pretty good predictor of where a team will finish – and also a decent yardstick to measure how well a team has done relative to their resources.

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Maths Studies IA Exploration Topics – A large number of examples of statistics investigations to explore.

In sports leagues around the world, managers are often only a few bad results away from the sack – but is this all down to a misunderstanding of statistics?

According to the Guardian, in the 21 year history of the Premier League, approximately 140 managers have been sacked. In more recent years the job is getting ever more precarious – 12 managers lost their jobs in 2013, and 20 managers in the top flight have been shown the door in the last 2 years. Indeed, there are now only three Premier League managers who have held their position for more than 2 years (Arsene Wenger, Sam Allardyce and Alan Pardew).

Owners appear attracted to the idea that a new manager can bring a sudden improvement in results – and indeed most casual observers of football would agree that new managers often seem to pull out some good initial results. But according to Dutch economist Dr Bas ter Weel this is just a case of regression to the mean – if a team has been underperforming relative to their abilities then over the long run we would expect them to improve to get closer to the mean value.

As the BBC reported:

*“Changing a manager during a crisis in the season does improve the results in the short term,” Dr Bas ter Weel says. “But this is a misleading statistic because not changing the manager would have had the same result.”*

*Ter Weel analysed managerial turnover across 18 seasons (1986-2004) of the Dutch premier division, the Eredivisie. As well as looking at what happened to teams who sacked their manager when the going got tough, he looked at those who had faced a similar slump in form but who stood by their boss to ride out the crisis.*

*He found that both groups faced a similar pattern of declines and improvements in form.*

By looking at the graph at the top of the page it is clear to see that sacking a manager may have appeared to lead to an improvement in results – but that actually had the manager not been sacked results would have been even better!

We can understand regression to the mean better by considering coin tosses as a crude model for football games (ignoring draws). If we get a head the team wins, if we get a tail the team loses. So this is a distinctly average team – which over a season we would expect to finish around mid-table. However over that season they will have “good runs” and “bad runs.”

This graphic above is the result of 38 coin tosses (the length of a Premier League season). Even though it’s the result of random throws you can see a run of 6 wins in a row – a good run. There’s also a run of 8 defeats and only 2 wins in 10 games – which would have more than a few Chairman thinking about getting a new manager.

Being aware of regression to the mean – i.e that over the long term results tend towards the mean would help owners to have greater confidence in riding out “bad runs” – and maybe would keep a few more managers in their jobs.

You can read the original research paper here.

**Premier League Finances – Debt and Wages**

This is a great article from the Guardian DataBlog analysing the finances for last season’s Premier League clubs. As the Guardian says, “More than two thirds of the Premier League’s record £2.4bn income in 2011-12 was paid out in wages, according to the most recently published accounts of all 20 clubs. The Guardian’s annual special report of Premier League clubs’ finances shows they spent £1.6bn on wages last season, most of it going to players.”

The first graph (above) shows the net debt levels for different clubs.

The second graph shows the total turnover:

The third graph shows wages as a proportion of turnover:

and the last one is particularly interesting – as it ranks clubs on their wage bills and their league position. This would be an interesting piece of data to test for the strength of correlation:

I’ve used an online scatter plot to calculate both the regression line and the correlation coefficient:

Which clearly shows a strong positive correlation. This would be an interesting exercise for both IGCSE or IB students (especially Maths Studies).

For even more data, a club by club full breakdown is also provided by the Guardian here. I have also made the data above into a word document to be used as a some A4 posters – and you can download that here: Premier League Debt

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