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A Primer: PDO and Normal Distributions

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Many of our readers don't have backgrounds in statistics. With the Leafs' recent skid after a coaching change I think some of this might be interesting.

John E. Sokolowski-USA TODAY Sports

Peter Horachek took over the Leafs and they promptly went 1-6 in his first seven games, failing to score in many of them. You may have heard people say that the Leafs haven't played that poorly, they've just had a poor PDO.

What's PDO?

PDO is a measure of "getting the bounces" in hockey. It's "puck luck". You add up a team's shooting percentage (under Horachek the Leafs have scored 8 goals on 198 shots for a 4.0% shooting percentage) and the goalies have stopped 90.2% of the shots they've faced (174/193=.902). Both of these numbers are well below league averages and both of these numbers are not likely to stay that low.

If we add those together we get 0.942, which we'd call a PDO of "942" (we throw away the 0 and the decimal to make it easier).

Why should I care about PDO?

A league average PDO would be 1000, because for every goal scored in the league one has to be given up. Over reasonable sample sizes (say half a season) PDO tends to regress towards 1000. "What if a team shoots better than another team?" and "What if my team has a great goalie?" seem to be good questions, but PDO regresses back towards 1000 nonetheless.

There's a great article about PDO at Arctic Ice Hockey available here. PDO is named after a commenter at Battle of Alberta named "PDO" who noticed that every time the Oilers traded a player away his PDO was really low, and every time they signed a guy or traded for a guy his PDO was really high. Oops.

In a post I'll write soon, I'm going to recreate some now lost knowledge wherein we break teams into six categories based on their PDO from the first quarter of the season. The top five PDO teams will be one bracket, the next five the next, etc. We would expect all six brackets to have PDOs closer to 1000 than where they started come the end of the season.

Why are we talking about PDO still?

First, this is a bit of a primer on PDO, and second the Leafs' PDO under Horachek is crummy. It's 940 and that's lower than 1000 which means it's bad. Unfortunately we might not have a lot of context for what a 940 means, so we're going to switch gears.

What's a normal distribution?

A normal distribution is sometimes called a Bell curve. It looks like this:

Normal Distribution

Image courtesy of Dan Kernler

A normal distribution tells us how data is spread out. If we take a bunch of random data that's normally distributed it'd look like that line above: most of it would be pretty close to the average, and there'd be a few outliers way far away.

What's a Standard Deviation?

A standard deviation is a measure of how the data is spread out. If the standard deviation is very low it means the data is all very close to the mean (commonly called the "average"). If the standard deviation is high it means the data is spread out from the mean.

So What Does Any of This Mean?

I took the PDO of every team's full season from 2002 until now, and made this chart:

PDO

It looks a lot like the perfect normal distribution up top. It's not perfect but it's ok. I also computed the standard deviation and the mean:

Mean 1000
Std. Dev 13.0

So looking at the black and white chart above we know a few things:

  • The Leafs' 940 PDO would be more than one standard deviation lower than the worst PDO ever in a full season. This isn't that weird: it's from only seven games and literally anything can happen.
  • The Leafs' 940 PDO under Horachek is 4.61 standard deviations below normal. 99.7% of all expected PDO values should be within 3 standard deviations, meaning from a PDO from 961 thru 1039 encompasses 99.7% of our PDOs in the big list.
  • That the Leafs have a 940 is incredibly rare. If they had a 940 over a full season we'd expect that to happen 0.00019% of the time, or once in 526,000 seasons, or once every 17,533 years.
Strong Conclusions

What all of this means is that the Leafs won't keep up a 940 PDO. Nobody is this bad, ever, and it's very difficult to judge a team's progress if none of the bounces ever go your way.

Do I know for sure that Horachek is a good coach? No I don't, but I wouldn't rush to judge him while the team gets historically bad bounces.

The good news? People from the Leafs front office are certainly aware of PDO although some might find it harder to bet on regression to the mean when it's their job on the line.