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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.

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.