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premier league regression

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.

IB Revision

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If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

Website Stats


IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

  1. Investigation essentials,
  2. Marking criteria guidance,
  3. 70 hand picked interesting topics
  4. Useful websites for use in the exploration,
  5. A student checklist for top marks
  6. Avoiding common student mistakes
  7. A selection of detailed exploration ideas
  8. Advice on using Geogebra, Desmos and Tracker.

Available to download here.

IB Revision Notes

IB Revision Notes

Full revision notes for both SL Analysis (60 pages) and HL Analysis (112 pages).  Beautifully written by an experienced IB Mathematics teacher, and of an exceptionally high quality.  Fully updated for the new syllabus.  A must for all Analysis students!

Available to download here.

IB HL Paper 3 Practice Questions (120 page pdf)

IB HL Paper 3 Practice Questions 

Seventeen full investigation questions – each one designed to last around 1 hour, and totaling around 40 pages and 600 marks worth of content.  There is also a fully typed up mark scheme.  Together this is around 120 pages of content.

Available to download here.

IB Exploration Modelling and Statistics Guide

IB Exploration Modelling and Statistics Guide

A 60 page pdf guide full of advice to help with modelling and statistics explorations – focusing in on non-calculator methods in order to show good understanding. Includes:

  1. Pearson’s Product: Height and arm span
  2. How to calculate standard deviation by hand
  3. Binomial investigation: ESP powers
  4. Paired t tests and 2 sample t tests: Reaction times
  5. Chi Squared: Efficiency of vaccines
  6. Spearman’s rank: Taste preference of cola
  7. Linear regression and log linearization.
  8. Quadratic regression and cubic regression.
  9. Exponential and trigonometric regression.

Available to download here.

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