## On the spread of PDO throughout a Premiership season

Inspired somewhat by this post from NHL Numbers (h/t to Ben Pugsley for the link) I think this is going to be a really useful reference post.

PDO is the sum of a teams shooting percentage (goals/shots on target) and its save percentage (saves/shots on target against). It treats each shot as having an equal chance of being scored – regardless of location, the shooter, or the identity or position of the ‘keeper and any defenders. Despite this obvious shortcoming it regresses heavily towards the mean – meaning that it has a large luck component. In fact, over the course of a Premiership season, the distance a teams PDO is from 1000 is ~60% luck.

A couple of years ago I showed that, come the end of a Premiership season, teams will have a PDO of 1000, with a standard deviation of 60. However, as with any metric, PDO has a larger spread over a small number of games than a full season. This has been illustrated this season, where the spread in PDO was 820-1120 after each team had played 9 Premiership games, but had dropped to 850-1110 after each had played 16. However it would be useful to know whether these are typical values, and also what we should expect the spread of PDO to look like when each team has played, say, 30 games of the season.

To determine that I took the last 13 Premiership seasons and calculated the PDO of each team after each game of the season. As a result I get the PDOs for each of 260 teams after each game of the season. For each game number I determined the standard deviation of the 260 PDOs. Statisticians typically use a measure of two standard deviations from the mean in order to decide whether or not a given result is statistically significant. As such, the lines on the plot below illustrate where the upper and lower bounds of statistical significance lie after a given number of games of the season.

To give this chart some context let’s consider a fictional team that recorded a PDO of 1200 in each and every game it played in a given season, and thus had a seasonal PDO of 1200 after each and every game it played. For the first 8 games of the season that’s perfectly normal – random bounces happen and over a small number of games (and thus a small number of goals) these can have a large effect on a teams PDO. After 9 games though a PDO of 1200 sits above the line of statistical significance. In other words it suggests that a team isn’t benefitting just from random bounces, but is also doing something skill based to push their PDO that high. For reference, as mentioned above not a single team had a PDO >1200 after 9 games of the ’13-14 Premiership season.

If a team managed to sustain a PDO of 1200 all of the way through the season they’d finish 3.6 standard deviations above the mean, and we’d be >99.9% sure their true PDO talent was above the mean of 1000. A full season PDO of 1200 simply doesn’t happen however, and the highest PDO a team has ever posted across a full season is the 1130 by Chelsea in ’04-05. This is still damn high though – it leaves us ~98% sure that their true PDO talent that season was >1000. It’s worth noting that the following season (’05-06) Chelsea’s PDO regressed significantly – down to 1087 (still high, but not significantly so).

However, let’s say you aren’t an especially good team, but instead you’re an average one. Exactly average to be precise. Your team takes and concedes the league average number of both shots and shots on target, and as far as we know is also league average at converting and stopping the shots on target in its matches. We’d expect this team to have a goal difference of 0 over the course of the season, and score a league average 52 points. Lets examine what happens to the goal difference and points* scored over a whole season by our average team if we simply vary its full season PDO from 900 to 1100 – values that still aren’t statistically significant over the course of a season.

*it’s demonstrated here that a goal is worth ~0.65 Premiership points for an average team.

So just by varying PDO within the not-statistically-significant range the average team scores 15 points more or fewer than 52 over the course of the season. Seriously – luck can have this large an effect upon the Premiership standings. It’s part of the reason we saw an average Newcastle team finish 5th in ’11-12, and also part of the reason we saw the same average Newcastle team finish 16th in ’12-13. Ditto a below average Reading team finishing 8th in ’06-07 and the same below average team finishing 18th a year later. Even within a season PDO can be both a kind and wicked mistress. In other words, if you see teams like Cardiff and Fulham only scoring a point per game early in the season despite having a high PDO you should be worried – their point scoring is probably due some regression.

Finally, to give some context to the PDOs we’ve seen thus far in the ’13-14 season I’ve taken the plot above and added each teams season long PDO after each of their games played thus far.

As expected the vast majority of dots fall within the limits of statistical significance. The few that wandered above in recent weeks were the values belonging to Arsenal, who’ve since regressed towards the mean. On the low end belong variously to Sunderland (week 8), Palace (weeks 10-14, and week 17) Norwich (week 10), and to Spurs (week 16). Those not named Palace have regressed significantly towards the mean since those points in time, and we should probably expect Palace to do so in the near future. Furthermore, I’d suggest it isn’t coincidence that three of those four teams have fired managers around the times they dropped below this line, and as such I’d say there’s likely an element of bad luck on the behalf of those managers.

Basically, after 17 games of the ’13-14 season only Palace have a PDO that is above/below that expected to a statistically significant extent. We should probably expect one team to end up above or below come the end of the season – but even picking Palace at this point wouldn’t be much more than a partially educated guess.