Pythagorean Expectation

I posted some stats to Twitter last night for Canada West women's hockey, and one of the statistics was something called "Goal Percentage". This is a fairly basic calculation that should give an idea how good teams are in comparison to one another when it comes to scoring goals in games, but it may show a few other things when analyzing the results in comparison to points and standings. What I was curious about was whether a value could be found for what a goal is worth in terms of points in the standings for each of the teams. The math might get a little hairy here, so let's look at last season's stats to be sure we're getting the full info.

CWUAA WOMEN'S HOCKEY

School	Record	Points	GF	GA	Streak	Next
Alberta	19-8-1-0	59	81	31	W1	BYE
Calgary	17-7-1-3	56	61	42	W2	BYE
Saskatchewan	14-7-3-4	52	55	42	L1	vs UBC
Mount Royal	12-14-2-0	40	43	44	W3	vs REG
Regina	9-13-5-1	38	48	54	W1	@ MRU
British Columbia	9-13-1-5	34	41	59	L2	@ SAS
Manitoba	8-17-2-1	29	34	63	L1	OUT
Lethbridge	7-16-2-3	28	48	76	L3	OUT

Goal percentage is a simple equation that figures out how many goals were scored by a team in all the games played out of the total number of goals scored in those games. The math looks like this: G% = GF/(GF+GA). This is a pretty simple statistic to figure as I stated above, so let's run through these numbers and look at the percentages.

1. Alberta: 59 points while scoring 81/112 = 72.32%
2. Calgary: 56 points while scoring 61/103 = 59.22%
3. Saskatchewan: 52 points while scoring 55/97 = 56.70%
4. Mount Royal: 40 points while scoring 43/87 = 49.43%
5. Regina: 38 points while scoring 48/102 = 47.06%
6. UBC: 34 points while scoring 41/100 = 41.00%
7. Manitoba: 29 points while scoring 34/97 = 35.05%
8. Lethbridge: 28 points while scoring 48/124 = 38.71%

If we look at Alberta's percentage, they were a full 13.1% better than Calgary, but only finished three points better in the standings. Clearly, there's something off there when it comes to Alberta's goal percentage in comparison to their point total, and it can be explained that Alberta blew out a few teams this year, ballooning that goal percentage. That's not good when it comes to finding a value of goals in terms of points because those blowout scores devalue the actual point value of the goals they scored.

Conversely, Manitoba finished one point ahead of Lethbridge in the standings despite scoring 3.66% less in their games than Lethbridge did. In knowing that, the goals that Manitoba scored as the lowest-scoring team in the conference would generate a high point value than what Lethbridge experienced with their goals. It could also suggest that Lethbridge was blown out by teams more often, contributing to a lower percentage than perhaps what they deserve, and that too is not good when finding the value of goals in terms of points earned per goal.

So why do we care about the goal percentage? Well, it is one of the ways we can predict standings. As you can see by the results above, the percentages follow a fairly predictable path from top to bottom with Alberta being the highest and Lethbridge being near the bottom. Is it accurate? Not entirely as there have been seasons where teams who finished lower in the standings had higher goal percentages, but it's fairly accurate when needing a quick look.

What I didn't post on Twitter because it needed some further exploring was that goals are directly tied to wins and points. Specifically, if we're looking at three-point possibilities for teams, we need to make some adjustments since shootouts don't factor into the goal-scoring or goals-against totals. Rather than assign a one- or two-point decision for shootouts and overtimes, we'll make them all differences of wins and losses - that is, two-point games are worth 0.67 wins and 0.33 losses while one-point games are worth 0.33 wins and 0.67 losses.

Alberta would finish with a record of 19.67 wins and 8.33 losses. Calgary would finish with an 18.67-9.33 record. Saskatchewan would have a 17.33-10.67 record. Mount Royal would move to a 13.33-14.67 record. Regina climbs to a 12.67-15.33 record. UBC ends up at 11.33-16.67. Manitoba improves to 9.67-18.33. Lethbridge finishes up at 9.33-18.67.

With each record now appropriately set based on wins and losses with extra wins and losses accounted for appropriately, we can now see an accurate display for wins and points and use the Pythagorean Expectation for predictions on both. To find this value, we simply square the numbers in the goal percentage formula for each team, and then multiply by 28 to get the expected wins and by 84 to get the expected points for each team. If the expected number is higher, the team underperformed OR had a number of lopsided games during the season, leading to increased expectations. If the expected number is lower, the team overperformed during the season OR were on the wrong side of blowouts, leading to lowered expectations. Make sense? Either way, we're going to take a look.

1. Alberta: .8722 expectation for 24.42 xWins and 73 xPts
2. Calgary: .6784 for 19.00 xWins and 57 xPts
3. Saskatchewan: .6317 for 17.69 xWins and 53 xPts
4. Mount Royal: .4885 for 13.68 xWins and 41 xPts
5. Regina: .4414 for 12.36 xWins and 37 xPts
6. UBC: .3256 for 9.12 xWins and 27 xPts
7. Manitoba: .2256 for 6.32 xWins and 19 xPts
8. Lethbridge: .2851 for 7.98 xWins and 24 xPts

It seems pretty clear that Alberta had a number of blowouts where they won big, leading to big expectations. The 14-point gap between their expected points and their actual points is a nearly unfathomable chasm, but it shows that the additional 4.75 wins they were expected to win with that offensive output wasn't realized. Would I say Alberta underperformed? They did win Canada West, but it seems the blowouts might have masked the fact that they earned close wins against teams that were better prepared for them.

Calgary's expected wins was .33 wins higher than their actual total, and the expected point total was exactly one point better than their realized point total. Calgary didn't underperform, but they were exactly as expected with their offensive and defensive efforts at the end of the season. The same goes for Saskatchewan and Mount Royal who both finished .33 wins below expectation and one point back of their expected point total, so they played as advertised.

We know that Regina went on a mad tear this past season where they rattled off something like eight-straight wins after starting the season ice-cold, and that's one of the reasons why this expected wins and points scenario needs some context. While Regina was .33 wins better than their expected wins total and one point better than their expected points total, the story of Regina was in those two streaks. The expected totals could have been so much better or worse had neither of those streaks happened.

The bottom of the standings is where things get a little wacky. UBC and Manitoba both played through extremely long scoreless droughts this season, leading their expectations to be incredibly lower than where they actually finished. UBC finished the season 2.21 wins above their expected total and 7 points better than their expected point total, so they actually exceeded expectations despite their struggles this season.

Manitoba, for their part, finished a clear 3.34 wins higher than their expected total and a full ten points better than their expected points total. Again, the math works in that 3.33 wins would be 10 points, but Manitoba's scoring woes might have been talked about a little too much when it came to what the stats were showing for an expected finish. Manitoba's actual totals were the highest above expectations for any team despite their scoring struggles this season!

And that leads us to Lethbridge who missed the playoffs and finished last yet again, but still exceeded expectations thanks to being on the wrong side of a few blowouts. They did score 48 goals this season - higher than Mount Royal, Manitoba, and UBC - but the goals-allowed total was off the charts as they were the recipients of a few hammerings. The expectations were lower, obviously, but Lethbridge actually performed better than expected as they finished 1.35 wins higher than expected and four points higher than expected.

So what does this all mean? Honestly, it's simply a way to determine whether teams played or are playing above or below their potentials with regards to their offensive and defensive efforts within a season. Alberta won Canada West again last season, so could they have raised their game to their expected levels? Possibly, but the end justified the means for them. And Mount Royal ousted two pretty good teams in Saskatchewan and Calgary to advance to the National Championship, so let's not take expectations as gospel when it comes to the results.

In the end, goal percentage and expected wins and points are simply ways to predict how teams might finish. Things could go incredibly wrong in the goal-scoring department as we saw with Manitoba, leading to a lower goal percentage and lower expectation than the team below them in Lethbridge. In a 28-game season, there may be wild swings in expected wins and goals based on a week-to-week basis thanks to goals scored and goals allowed, and that's another reason why these expectations and predictions shouldn't be taken as certain things.

I do find it interesting that the math works out in both cases where the goal percentages yielded an almost perfect replica of the standings while the expected wins and points had a few quirks shown thanks to scoreless streaks and lopsided scores. Math can and should be fun, and this little exercise was a bit of fun I had in the middle of the week while I wait for Canada West hockey to return.

Until next time, keep your sticks on the ice!