Hypergeometric statistics are fun

I got curious the other day to see just what the odds of the recent 6-0-0 run were. More specifically, is it possible that SKC is just an average team who have gotten lucky? In the end, of course, it depends on how you define luck, but the math is pretty easy to work out. Assuming an "average" team plays like a 3 sided coin and wins 1/3rd of the time, draws 1/3rd of the time, and loses 1/3rd of a time, it's pretty easy to see that the odds of six in a row is (1/3)^6, 0.0014, or 0.14%. That's really small. It would be expected to happen only once every 714 runs of six games, assuming they're truly an average team. Of course you could have a run of six games every time. Assuming a 36 game season, this would happen about once every 20 years. Since the league has been around 16 years, it would stand to reason that over the course of the league's history, this would happen about once. That it's happened twice now (LA in '96 did it first) isn't really all that surprising when looking at the league as a whole. But really, for any team to do it is pretty remarkable, and I think it's much more likely to be the case that SKC isn't just lucky, but good.

To go a little further, he's a breakdown of how the league would look with true parity (i.e. each team wins 1/3rd of the time, draws 1/3rd of the time and loses 1/3rd of the time).

Points	%	Expected # of teams	Full Teams
0	0.14%	0.03	0
1	0.82%	0.16	0
2	2.06%	0.39	0
3	3.57%	0.68	1
4	6.17%	1.17	1
5	9.05%	1.72	2
6	10.42%	1.98	2
7	12.35%	2.35	2
8	13.17%	2.50	3
9	10.97%	2.09	2
10	10.29%	1.95	2
11	8.23%	1.56	2
12	4.80%	0.91	1
13	4.12%	0.78	1
14	2.06%	0.39	0
15	0.82%	0.16	0
16	0.82%	0.16	0
17	0.00%	0.00	0
18	0.14%	0.03	0
sum	100.00%	19.00	19

 

See that blip on the far right? That's SKC. Now, obviously we don't have fractional teams, so here's how we'd look with full teams only.

Pretty boring, huh?

Well, that's all fine and good, but I decided to try to play with the percentages to see what win-draw-loss rates would give SKC a reasonable chance to go 6-0-0. The percentrages all feed in to the hypergeometric distribution as such.

 Initially I used 1/3rd for each probability, but adjusting the wins % lets you play with the odds. To go from 0.14% to a reasonable probability, in this case I chose 5% (i.e. there's a 20-1 chance that SKC goes 6-0-0), I needed to adjust the win proability from 33% to 60.7%. I left draw and loss probabilities equal to each other, so they came out as 19.65% each.

Basically, what it looks like from six games so far is that Kansas City should win about 60% of their matches, and not win the remaining 40%. If we break it up equal as draws and losses we can then come up with some predicted points. Let's look at the conservative case first, in which they win 60% and lose 40%.

Based on where they are now, with 18 points and 28 matches to go, assuming they win 60% of the remaining matches (28 *0.6) would give them 16.8 wins. Let's round down to 16 and say they earn 48 more points, putting them at 66 points.

In 2011, LA won the shield (and the MLS Cup) on 67 points, and some took to asking if they were the best MLS team ever.

Now let's add draws in. 16 wins out of 28 means 12 not wins. Split those evening and we get 6 draws, or 6 more points. So, 6 + 67 would put SKC on 73 points at the end of the season.

To be clear, I don't neccessarily think SKC is a 60%-20%-20% over the course of the season, but I think they're definitely playing like it right now. With the road trip to Vancouver and Portland on short rest this week we might to start seeing some early injuries, and summer months in the US can really beat on a team physically. Their depth is going to be tested, for sure, but right now, SKC looks to be on pace to flirt with a 70 points season.

In short, I don't think this is some kind of statistical fluke. After beating RSL last week, I really think this team is the real deal.