Monday, January 31, 2011

Inside The Numbers: Beware Variance

With 14 games down, the league has begun to sort itself out a little bit. There are still three tiers, but the tiers are a lot more even than when the bottom and top combined for just three teams with five left in the middle.


Harvard           22             6           11             3
Princeton           23             7           11             3
Yale           15           13             8             6
Columbia           17           11             8             6
Penn           14           14             8             6
Cornell             8           20             4           10
Brown           11           17             4           10
Dartmouth             7           21             3           11

The Crimson and Tigers retain their stranglehold on the top two spots, especially now that the Bulldogs have fallen further back to the pack. The Lions and Quakers survived early tests to position themselves neatly behind the leaders within striking distance of the top of the Ivies if they can pull off a select upset or two.

After getting swept by their travel partners, the title hopes were shot anyway, but with the Big Red losing at Dartmouth and the Bears getting swept again as well, Cornell and Brown join the Big Green in a pack that will likely be jostling for a top prize of sixth.

OFFENSIVE LEADERS (Avg. ORating, Standard Deviation - Rank)

1. Harvard (105, 12.6 - 4th)
2. Princeton (104, 8.7 - 8th)
3. Columbia (101, 10.3 - 7th)
4. Penn (99, 11.5 - 6th)
5. Cornell (98, 13.5 - 2nd)
6. Brown (97, 11.6 - 5th)
7. Yale (97, 14.2 - 1st)
8. Dartmouth (86, 13.5 - 3rd)

The Crimson continues to pace the league in offensive rating, scoring over 105 points per 100 possessions, just ahead of the Tigers. The standard deviation becomes important in the expectation of outcomes. Princeton has been a lot more consistent offensively this year (most consistent in the league by a decent margin), which should give the Tigers a leg up in a format where consistency is highly valued.

Dartmouth brings up the rear - this will be the eighth consecutive year that the Big Green has finished 7th or 8th in ORating - but has been inconsistent enough to throw in some surprisingly good performances that have led to interesting victories.

DEFENSIVE LEADERS (Avg. DRating, Standard Deviation - Rank)

1. Yale (97, 14.2 - 1st)
2. Princeton (98, 10.4 - 7th)
3. Harvard (98, 10.9 - 5th)
4. Dartmouth (102, 7.9 - 8th)
5. Penn (103, 11.8 - 3rd)
6. Cornell (104, 11.4 - 4th)
7. Brown (104, 11.9 - 2nd)
8. Columbia (105, 10.7 - 6th)

Why has Yale been up and down all year? Look no further than the variance numbers. The league's best defense is also, by far, its least consistent, and that's most likely because its best defensive player, Greg Mangano, picks up 4.8 fouls per 40 minutes and spends a decent amount of time in foul trouble. Despite the fact that its offensive and defensive ratings combine for a nationally average team, the high variance performances on both ends of the floor produce a team that can play with the best and lose to the worst.

Once again, Harvard and Princeton find themselves among the leaders, but the Tigers have been much more consistent. That should benefit them over a 14-game stretch.

Penn and Columbia each fancy themselves as contenders, but must close the gap between themselves and the leaders on the defensive end to sustain the run late into February.


The mean and variance are key to understanding a team's quality and what you might be able to expect on any given night. There's one more piece of the puzzle, however. It's called co-variance. In basketball terms, it measures any correlation there might be between a team's offensive and defensive performances by game.

Theoretically, the co-variance should be zero. How you play offensively should have nothing to do with how you play defensively, although in practice, that is rarely the case, and those with a more nuanced understanding of basketball could probably generate hundreds of theories and examples as to why.

The importance of this metric can be illustrated by Yale. The Bulldogs are the league's highest variance team on both offense and defense. Their win/loss margins, however, are among the least variable in the league. This apparent paradox is resolved by co-variance. Yale's co-variance is highly positive - the highest in the league over the past two years - meaning that as the team plays better offense, it plays worse defense and vice versa. Thus, the results in Yale's games remain relatively consistent (not the total points but the margins), because any time the Bulldogs do something well at one end of the court, they tend to do something poorly at the other end.

Among the other Ivy teams, Cornell also has a highly positive co-variance, while Brown, Penn and Princeton are positive, but not all that significant. Harvard, Columbia and Dartmouth all hover around zero, meaning that those teams' performances on each end of the floor are pretty much uncorrelated with each other.

REBOUNDING MARGIN (OReb% + DReb% - 100%)

1. Columbia: 8%
2. Princeton: 2%
3. Harvard: 2%
4. Yale: -3%
5. Brown: -4%
6. Cornell: -5%
7. Penn: -6%
8. Dartmouth: -8%

The league-leading Lions struggled this past weekend, getting solidly out-rebounded by Harvard (the worst rebounding margin for Columbia all season) and barely getting the better of the battle of the boards against Dartmouth, though much of that could be attributed to missing Brian Grimes and having Asenso Ampim on the bench for most of the weekend.

While Yale, Brown and Penn have negative margins, their defensive rebounding has been very good (all Top 100). It's the offensive rebounding that is tripping those teams up, potentially because some of the best offensive rebounders on these teams are also good enough three-point shooters that they're often on the perimeter and not inside to gather in wayward shots.

95% CONFIDENCE INTERVALS (Mean Ivy Wins, 95% High, 95% Low)

1. Harvard (10.9, 14, 8)
2. Princeton (10.5, 13, 8)
3. Yale (8.0, 11, 5)
4. Columbia (7.6, 10, 5)
5. Penn (7.5, 11, 5)
6. Cornell (4.3, 7, 2)
7. Brown (4.3, 7, 2)
8. Dartmouth (2.8, 5, 1)

Bringing this piece back full circle, if you want evidence of why variance matters, take a look at Penn. The Quakers are slightly behind Columbia in win expectations but have a slightly higher ceiling, due to the fact that the team has higher offensive and defensive variance game-to-game. It's also the reason why Penn has a 3.4% chance of winning the title, while the Lions are at 2.4%.

Yale's title hopes took a dive this weekend, falling from double-digits to just 4.0%. Harvard and Princeton are the prohibitive favorites at this point, sitting at 49.7% and 40.5%, respectively.


1. Princeton 4-1
2. Harvard 5-1
3. Columbia 4-2
4. Yale 4-2
5. Penn 3-2
6. Brown 1-5
7. Dartmouth 1-5
8. Cornell 1-5

NOTE: By expected winning percentage, Penn actually ranks 3rd coming out of next Tuesday's meeting with Princeton. The Quakers' expected 3.4-1.6 record (.673) rounds down to 3-2, while the Lions' 3.7-2.3 record (.611) and Bulldogs' 3.5-2.5 record (.589) round up.

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