We’re in the thick of the Stanley Cup playoffs, which means hockey fans are doing what seems to be every sports fan's favorite hobby...complaining about the refs! While most complaints, such as “We’re not getting any of the close calls!” are subjective and hard to get data for, there's one question that we should be able to answer objectively with a statistical analysis: Are hockey penalties independent trials? That is, does the team that the next penalty will be called on depend on the team that any previous penalties were called on?

Think of flipping a coin. Even if it comes up heads 10 times in a row, the probability of getting heads on the next flip is still 50%. In theory, you would think penalties in hockey work the same way. Both teams are playing hard, and should be equally likely to commit the next penalty. Maybe a single player would be less likely to commit a second penalty right after he just committed one because he’ll play more cautiously. But at the team level, you would think the outcome of the next penalty to be 50/50.

But players aren’t the only ones who affect the outcome of a penalty. Referees are ultimately the people who decide when to let things go and when to call a penalty. And you can only imagine what the crowd and coaches would do to the ref if the home team had 10 penalties called against them in a row.

So let’s dig into the data with Minitab Statistical Software and see if refs call penalties independently, or if the team they call it on depends on which team they called previous penalties on.

## The Data

I’m only going to include playoff games in my sample, because those are the games where the stakes are the highest. For every Stanley Cup Playoff game from 2013 and an additional 21 games from this year, I collected the team the penalty was called on (either home or away) and the order in which they were called. I only included penalties where one team got a power play. So if matching penalties were assessed to players on opposite teams, I didn’t include those since it didn't give either team an advantage. I also didn’t include penalties for fights that occurred late in hockey games that were blowouts. (By that point the game was effectively over, so the penalties didn’t matter.) In total I had 732 penalties that occurred in 106 playoff hockey games.

## The First Penalty

No penalties have been called at this point, so there shouldn’t be any bias. We should expect an equal amount of penalties to be called on the home and away team.

Sure enough the penalties are just about 50/50. So far, so good, refs!

## The 2nd and 3rd goals

Now let’s see what happens with the next two penalties, starting with the second. Is the team that the second penalty is called on independent of the team the first was called on? We can perform a chi-square test of association to get an answer.

Looking at the table, we see that of the 51 times the first penalty was called on the away team, the next penalty was called on the home team 33 times (65%). And of the 55 times the first penalty was called on the home team, the next penalty was called on the away team 32 times (58%). It appears that the second penalty depends on the first, but we should examine the results of the chi-square tests to be certain.

The p-values for both chi-square tests are 0.018. Because this value is less than 0.05, we can conclude that there is an association between the first and second penalty. So if your team gets called for the first penalty, odds are the next one is going against the other team.

Will this trend continue for the third penalty? Let’s start by thinking about the number of penalties called on the home team. There could be 0, 1, 2, or 3 penalties called on them. If the penalties are independent of each other, then the probability of a single penalty being called on the home team at any point in time is 0.5. We can use this to easily calculate the probabilities for the different amount of penalties that could be called on the home team.

 Penalties on the home team Equation Probability 0 .53 0.125 1 3*(.5)*(.52) 0.375 2 3*(.52)*(.5) 0.375 3 .53 0.125

Now we just have to summarize our data, and see how many times each of these actually occurred. Then we can use a chi-square goodness-of-fit test to compare our observed values to the expected probabilities that we calculated above.

If you look at N in the bottom left corner, you’ll see that our sample size dropped to 103 games. That’s because there were 3 games where only 2 penalties were called, so they couldn’t be included in this analysis.

Now let’s focus on the table. In a sample of 103 games, we would expect there to be about 13 games where the home team had 0 penalties and another 13 where they had 3. But the Observed column shows us that there were far fewer. In fact, there were only 3 games where the first 3 penalties all went to the home team. It seems like the refs were reluctant to get the home crowd angry at them.

The 1 and 2 penalty categories suggest that the refs appear reluctant to have anybody get mad at them. While we would expect about 39 games to occur in each category, there were 46 and 47 instead!

The p-value for the chi-square test is 0.004. This means we can conclude that the data do not follow the proportions we would expect if the trials were independent. When it comes to the first 3 penalties of the game, the refs are reluctant to give either team too much of an advantage (especially the away team), instead opting to make the penalties as even as possible.

## The Entire Game

Now let’s move on from the first couple goals and try to determine what happens throughout the entire hockey game. For every penalty throughout a single game, I gave it a 1 if it was called on the home team and a -1 if it was called on the away team. Then I added these values up throughout the game to keep track of the “count”. So if the first 3 penalties were called on the away team, the count is at -3. And if 3 of the first 5 penalties were called on the home team, the count would be at 1. Here is a histogram of the counts.

Notice that it is very rare for the count to move too far from 0. Counts over 2 or less than -2 are pretty rare, once again showing that refs don’t want to seem too biased toward either team. Because counts of 3 or higher (or -3 and lower) were rare, these samples sizes are too small to obtain any conclusions from. So I combined them with the 2 and -2 category to increase the sample size.

Once I had the count for every penalty in the game, I recorded the team the next penalty was called on. Then for each count, we can see the proportions for the team the next penalty was called on!

Let’s start with the negative counts. This means the away team has had more penalties called on them than the home team. We see that in these cases, the home team is slightly more likely to be called for the next penalty, but the probabilities are still close to 50%. Any bias the refs show in calling penalties does not favor the away team.

It’s quite a different story when we look at the home team. When the home team has been called for 2 or more penalties than the away team, the next penalty goes on the away team 75% of the time! And in case you think this is just a result of combining all the counts of 2 or more, I can tell you that if you take a count of 2 as its own category, the percentage of the next penalty going to the away team was still 74.65%.

So what is causing this? It’s unlikely to be the players or the coach of the home team yelling at the ref. After all, I’m sure the players and coach of the away team yell at the refs just as much when the count is negative, and they don’t see an advantage.

That brings us back to the fans. Remember when I said every sports fans favorite hobby seems to be complaining about the refs? Well, refs are humans, too, and it definitely seems plausible that when most of the calls are going against the home team, the crowd noise might cause the refs to be more prone to call the next penalty on the away team. Of course, this analysis can’t prove that this is the cause, but it adds some credence to a nice theory!

So if you ever find yourself at a playoff hockey game, feel free to boo as loud as you want when the ref calls a penalty on your team. Even if it was a good call!  You just might be tipping the odds in your team’s favor!