Tuesday, 8 August 2017

"It's All About The Distribution".

You've got five teams.

One is consistently the best team, their recruitment is spot on with a steady stream of younger replacements ready and able to take over when their starts peak and wane.

Then we've got two slightly inferior challengers, again the model of consistency, with few surprises, either good or bad.

The lowest two rated teams complete the group of five.

The marginally superior of these also turns in performances that only waver slightly from their baseline average.

For the final team, however we have very limited information about their abilities, partly due to a constantly changing line up and new acquisitions.

The current team has been assembled from a variety of unfashionable leagues and results and we only have a handful of results by which to judge them.

So we group together the initial results of similarly, newly assembled teams to create a larger sample size to describe what we might get from such a team.

Instead of a distribution that resembles the four, more established teams, we get one that is much more inconsistent. Some such teams did well, others very badly.

The distribution of performances for the first four sides is typical of teams from this mini league, whereas the distribution we have chosen to represent the potential upside and downside of this unexposed side is not.

Team 5's distribution has a flatter peak and fatter tails, both good and bad.

The average "ratings" of the five teams are shown below.

Team 5 has the lowest average rating, but by far the largest standard deviation based on the individual ratings of the particular cohort of sides we have chosen to represent them.

As Team 5 is the lowest rated, they're obviously going to finish bottom of the table, a lot, but just to confirm things we could run a simulation based on the distribution of performances for all five teams.

First we need to produce a distribution that mimics the range of performances for the 5 teams and we'll draw a random number from that distribution to decide the outcome of a series of contests.

The highest performance number drawn takes the spoils.

Run 10,000 simulated contests and Team 5 does come last more frequently than any other side, roughly half the tournaments finish with Team 5 in last position.

However, because their profiled performances are inconsistent and populated by a few very good performances, they actually come first more frequently than might be expected from their average performance rating.

In 10,000 simulations, Team 5 comes first 22% of the time, bettered only by Team 1, whose random draw of ratings based on their more conventional distribution of potential performances grants them victory 36% of the time.

Not really what you'd expect simply from eyeballing the raw ratings.

Team 5, based on the accumulated record of teams that have similar limited data, are likely to be sometimes very bad, but occasionally they can produce excellent results.

Such as Leicester when they were transitioning into a title winning team?

As someone once said at an OptaProForum.......

"It's all about the distribution"

......and simple averages can sometimes miss sub populations that could be almost anything.

Straight line assumptions, extrapolated from mere averages will always omit the inevitable uncertainty that surrounds such teams or players, where data is scarce and distribution tails might be fatter than normal.

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