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Regressing to the mean

A deep dive into what regressing to the mean actually means for the Bruins this season

NHL: Boston Bruins at Buffalo Sabres Timothy T. Ludwig-USA TODAY Sports

This is a “mathy” article, so there will be a summary at the bottom for those only interested in the key points of the article. Data via and 5v5 unless stated otherwise.


At the time that this is being written, the Bruins sit atop the Atlantic Division with a 24-7-11 record. They are second in the league at 5v5 goal share, and have strong special teams, giving them the best goal differential in the league at +32. Everything is good, right?

As Lee Corso says, “Not so fast, my friends.” There are some underlying concerns that seem to be present with the 5v5 offense. The chart below shows how the Bruins derive their goal differential. The shooting variable is colored green and makes up most of the Bruins 5v5 goal differential.

Personally, I think this is a better way of looking at a team’s PDO, for those familiar with that statistic. The shooting variable is just goals minus expected goals, which is 0.57 per 60 for the Bruins. The reason this is concerning is that the Bruins are 30th in the league at 5v5 expected goals per 60. If the Bruins were to regress all of the way to the mean, holding their goaltending where it is, the Bruins 5v5 goal differential per 60 would fall from 0.68 (2nd) to 0.11 (T-12 with three teams).

Now the Bruins would have to shoot miserably in order to regress all of the way to the mean, and that’s not exactly how regression to the mean works. Regression to the mean is just an easy way of saying that extreme results probably won’t continue. If you flipped a coin on heads four times in a row, it is not more likely that you will flip tails the next time you flip a coin. However, if you were to flip a coin four more times, it would be unlikely that you flip all heads again.

Additionally, who cares about their goal differential at the end of the season? The Bruins have a lot of points already in the bag and there is no reason to believe that they will fall off the face of the earth and miss the playoffs, or even end up in a wild card spot. What we really care about is knowing what to expect from the team moving forward and into the playoffs. The relevant question is, what is the Bruins “real” shooting percentage?

Getting down to business

For this exercise, I will ignore expected goals to an extent. For one, we already used them as a signal. Additionally, sometimes it is nice to add a bit of diversity and look at the numbers a bit differently. I will use a beta distribution to estimate the Bruins “real” fenwick (unblocked shots) shooting percentage.

The advantage of a beta distribution is that we can include a prior. In layman’s terms, we can use previous data to help fit this distribution better. Taking this to an extreme, if there was a team shooting 20% and we didn’t include any previous data, the computer would think this is normal. Luckily, we have lots of data on teams’ shooting percentages and have a good idea of what these look like.

With this data, we will find an alpha and beta. There are a few different ways to do this. You could do out the math through the method-of-moments as shown below. You could also use the package fitdistrplus in R, or the fitdistr function in the MASS package in R. For non-R users, you just have to google it.

For this distribution, we are making our prior the average NHL team. I took every team’s regular season fenwick shooting percentage since 2007-08 and dropped the shortened season. Below is my fitted distribution plotted.

The Bruins currently have a 6.47% fenwick shooting percentage at 5v5. That is not too wild as it is only 9th in the league, and only slightly above the league average of 5.97%. We can update our alpha and beta to get an idea of where the Bruins really stand by adding in the goals and unblocked shots up to this date, and plot the prior and posterior distributions.

If we believed the Bruins were an average shooting team at the beginning of the season, we would now believe that the Bruins are likely a slightly above average shooting team. The prior empirical fenwick shooting percentage was 5.66%, and now it is 5.84% and with much more certainty.

When looking at the graph, the probability density at 6.47% is quite low, but it’s more likely than it was to begin the year. However, if you were to draw a clean slate now and were asked what their fenwick shooting percentage would be for the rest of the season, between 5.60% and 6.00% would probably be your best bet, given this distribution.

There are a few pieces of information missing, of course. For one, shooting percentages have been up over the last couple of years. That could be random variance or a part of the rule changes, but all of the data going back to 2007-08 is weighed the same. Additionally, we are ignoring shot quality, or at least the shot quality we can attempt to capture using expected goals models. The Bruins have an expected fenwick shooting percentage far below average at 5.07% this season.

However, so far we have used a prior of average, but are the Bruins an above average shooting team? Last season the Bruins finished 26th in shooting percentage (shots on goal and not fenwick in this case). With the same coach and practically the same roster, shouldn’t we have believed they were a below average shooting team heading into the season?

We could try to repeat the same exercise using teams around the Bruins fenwick shooting percentage last season. I decided to pick teams +/- 0.25% of the Bruins last season. There were 86 teams that fit that, and because we are picking a somewhat random point on the overall distribution, the fit is a bit funky.

And quite frankly, it is overfit. The alpha and beta are quite large, so it takes a lot to move the posterior.

Again, given the new information we have from this season, one would be wise to believe the Bruins are a better shooting team than they did on October 1, but this is an extremely conservative move. The empirical estimate for the Bruins here is 5.32%.

So we have three estimates for the Bruins “real” shooting talent. We have an expected goals model and two empirical estimations from beta distributions. How many goals would the Bruins lose if we applied these estimates and backtracked?

Goals by model

Model FSh% Unblocked Shots Goals Difference
Model FSh% Unblocked Shots Goals Difference
None 6.47 1344 87 0
Expected Goals 5.07 1344 68 -19
Average Prior 5.84 1344 78 -9
Last Season Prior 5.32 1344 72 -15

I think realistically, the Bruins have scored around 10 to 15 more goals than their “real” talent. I think assuming they were an average shooting team to begin the year would be a bit optimistic, but the expected goals model is also likely a bit low on the Bruins.

Luckily for the Bruins, the goals are already in the bag, but things might need to change going forward. Sitting 30th in expected goals 60 and 19th in shot attempts per 60 is a bit concerning. The Bruins have already began to cool off a bit, and this hasn’t been a fun stretch of 15 games.


  • The Bruins are relying heavily on their shooting percentage at 5v5
  • Based on a few models, the Bruins’ true shooting talent is probably below what they are shooting right now
  • If the Bruins continue to take shot attempts at the same rate, their offense will likely dry up.