# Warning: there be math here

Well now, it’s been quite some time since I posted in my own blog (rather than the ECAC Hockey blog)… but I was bit by the curiosity bug, and it led me to do some quick research on D-I hockey as a whole rather than strictly the ECAC, so I figured it was time to resurrect this sucker.

The question on my mind, specifically, was about league “tightness”: a lot of coaches talk about how competitive their own leagues are, how their conferences are – top to bottom – the most talented or challenging or whathaveyou. How tight their leagues are, is what you’ll hear most often. So I did some mathematical querying to find out which leagues really are the tightest at the wire.

Here’s what I found. (If math scares you, consider yourself warned. Nerds: rejoice, for there be numbers on t’other side of yonder headline!)

#### And the Pepto Prize goes to…

…no surprise to me or my readers: ECAC Hockey. Try to shoot bias-sized holes through my methodology all you like, but I simply plugged the numbers and reported what spat out.

ECAC Hockey’s final regular-season standings have had the lowest standard deviation (symbolized by sigma, “σ”) score of any league in three of the last four years. To translate for the non-stat-minded, the lower the score, the tighter the league as a whole. One σ from the mean (average) will by definition cover 34.1 percent of the population… so if σ for a league is 8.0, for example, then over 68 percent of the league’s points (and likely, teams) will fall within eight points of the average.

This year, σ for ECAC Hockey – albeit with four games left apiece – is a measly 4.75. Meanwhile on the other end of the spectrum, Atlantic Hockey’s is 8.91; Hockey East’s, 7.29; CCHA’s, 7.22; and the WCHA’s, 6.05. That doesn’t mean there aren’t exciting races going on in those leagues… it just means that there is likely to be more at stake for more teams in the tighter conferences right now.

#### σ in recent history

The largest σ in the last four years was the CCHA’s 12.99 last year. Yawn. The tightest at the end of the regular season was ECAC Hockey’s 5.91 in 2007-08 (narrowly edging AHA’s 5.93 that same year).

The ECAC has had the lowest average σ over the past four (nearly five) years, as mentioned, with a 6.30 score. Here are the other leagues’ average σ over the same period:

• Atlantic Hockey: 8.30
• CCHA: 10.62
• The long-lost CHA: 7.55
• Hockey East: 8.36
• WCHA: 8.4

I imagine a few of you will find this interesting, while many will absolutely not. It’s not meant to say that any league is any better than any other. All it implies is that there are likely to be more and/or tighter races for postseason position in some leagues than in others, and that’s really all I set out to find.

1. In your calculations do you take into account the number of points per game that are available? For example the sigma of 7.22 points in the CCHA represents 2.4 times the points available in a game.  The ECAC sigma of 4.75 represents 2.37 times the points available in a game.  Not as big a difference when you look at it that way. ( A more standard way to look at it would be sigma/mean.  CCHA = 7.22/36.5 = 0.20     ECAC = 4.75/18 = 0.26   which makes the CCHA even tighter than the ECAC. But if you looking for how things might swing for the remaining games the sigma/game points is probably better. )

•  That’s a very good point: I neglected to appreciate that there are more points available per game in the CCHA due to shootouts. That said, I wonder if such adjustments would be necessary, as more points on the table doesn’t necessarily warp the ratios between the good and bad teams…?

• Yes, but the standard deviation is NOT a ratio.  It represents the width of the distribution of, in this case, the number of points that each team has gotten so far in the season.  When you calculate a standard deviation of a set of numbers you’re assuming that the numbers are distributed around a mean value with a normal or Gaussian distribution.  That’s where the 68% number comes from.  Rescale the CCHA points by 2/3 (as if each game counted for 2 points) and the standard deviation becomes 4.8.

(Remember that the CCHA is not like the NHL – you don’t just get an extra point for winning a shootout.  It’s a 3/2/1 scale for Win/Shootout Win/Shootout Loss, so the direct 2/3 scaling is appropriate.)

2. In your calculations do you take into account the number of points per game that are available? For example the sigma of 7.22 points in the CCHA represents 2.4 times the points available in a game.  The ECAC sigma of 4.75 represents 2.37 times the points available in a game.  Not as big a difference when you look at it that way. ( A more standard way to look at it would be sigma/mean.  CCHA = 7.22/36.5 = 0.20     ECAC = 4.75/18 = 0.26   which makes the CCHA even tighter than the ECAC. But if you looking for how things might swing for the remaining games the sigma/game points is probably better. )

•  That’s a very good point: I neglected to appreciate that there are more points available per game in the CCHA due to shootouts. That said, I wonder if such adjustments would be necessary, as more points on the table doesn’t necessarily warp the ratios between the good and bad teams…?

• Yes, but the standard deviation is NOT a ratio.  It represents the width of the distribution of, in this case, the number of points that each team has gotten so far in the season.  When you calculate a standard deviation of a set of numbers you’re assuming that the numbers are distributed around a mean value with a normal or Gaussian distribution.  That’s where the 68% number comes from.  Rescale the CCHA points by 2/3 (as if each game counted for 2 points) and the standard deviation becomes 4.8.

(Remember that the CCHA is not like the NHL – you don’t just get an extra point for winning a shootout.  It’s a 3/2/1 scale for Win/Shootout Win/Shootout Loss, so the direct 2/3 scaling is appropriate.)