Right now I'm watching the Michigan vs. Florida game in the Elite 8 of the NCAA tournament. The winner of this game will advance to one of the coveted Final Four positions, and have a chance of winning the National championship.

By most measures Michigan totally dominated Florida in the first half. Florida has never led. Michigan scored the first 13 points of the game. At one point, Michigan led by 24 points.

However, I noticed something that I see over and over again in basketball games: total blowouts are extremely rare. Once Florida scored its first points, the maximum fractional lead was 13 points divided by 2 points, or a factor of 6.5 times as many points as Florida. What I find puzzling is that this fractional lead steadily decreased steadily as the game advanced toward halftime.

This is puzzling because let's suppose that Michigan is 1.5 times better than Florida (on this specific day). This factor is likely smaller given that the teams are ranked 4 and 3, respectively. But let's just suppose Michigan (or any other basketball team) is 1.5 times better than their opponent. This would mean that on any given possession, they have a factor of 1.5 larger chance of hitting their shots. On defense, they should be 1.5 times more likely to force a turnover.

Thus, if Florida has a 30% chance of scoring on any given possession, Michigan should have a 45% chance of scoring. If this is true, shouldn't Michigan's score with 15 minutes left in the half be a factor of 1.5 times larger than Florida's at that point? If Florida scored 16 points with 15 minutes left, then Michigan should have 24 points. Similarly, at the end of the half, Michigan should have 1.5 times more points. The fractional score should be constant throughout the game, right?!

However, check out the fractional score (ScoreM/ScoreF) as a function of time:

(Note: Since Florida didn't score until the 16:29 mark, the ratio is undefined for the first 3.5 minutes)

While Michigan led throughout the first half, their fractional lead decreased almost monotonically, from 6ish down to 1.5.

In games with mismatched teams, I see this happen all the time. The better team jumps out to an early lead that is huge in a fractional sense. Something like 20-5, or a factor of 4 lead. But thereafter the score difference typically remains roughly constant at, say, 15-20 points. But the fractional lead decreases steadily throughout the game. One team clearly dominates, but they very rarely pull all the way away. This is especially true in NBA games and NCAA tourney games (based on my casual observation).

If we define a true blowout as a game during which one team maintains a nearly constant fractional lead over their opponent (with a final score something like 90-50), why are blowouts so rare? I can think of a few reasons, but none are particularly satisfying:
1. Teams let up when they have a big lead, putting in bench players who then match up against the opposing (losing) starters (this is the most compelling hypothesis IMO).
2. The losing team tries harder when down by a large amount (doubtful. I'd expect the opposite).
3. The early portions of games are dominated by small-number statistics, and as the game goes on, the fractional lead asymptotically approaches the score expected based on the fractional disparity in ability between the two teams. (this would be interesting, but I don't understand why it would work out this way).

BTW, there are 5 minutes left in the game, and the score is 69-51 Michigan, or a 1.35 fractional score. The fractional score has continued to decrease after the half! If the fractional score 15 minutes into the first half (2) held, then the score would be 69-35 right now. But an Elite Eight game with that sort of score difference (34 point differential) is almost unheard of. Why?!

TJR said…
I'm a big NBA fan (NY Knicks) and see this all the time. I don't think bench vs. starters is really the answer, at least in the NBA, because coaches usually have set rotations (e.g., bench players typically start the 2nd and 4th quarters), so it's usually bench vs. bench and starters vs. starters.

I personally think it's more of a psychological factor. If you're up big, you naturally try to conserve energy by exerting less effort because you have more room for error. This happens to me when I'm playing pick-up and my team is up big (doesn't happen that often though!).

-Timothy Rodigas (U. Arizona astro)

### On the Height of J.J. Barea

Dallas Mavericks point guard J.J. Barea standing between two very tall people (from: Picassa user photoasisphoto).

Congrats to the Dallas Mavericks, who beat the Miami Heat tonight in game six to win the NBA championship.

Okay, with that out of the way, just how tall is the busy-footed Maverick point guard J.J. Barea? He's listed as 6-foot on NBA.com, but no one, not even the sports casters, believes that he can possibly be that tall. He looks like a super-fast Hobbit out there. But could that just be relative scaling, with him standing next to a bunch of extremely tall people? People on Yahoo! Answers think so---I know because I've been Google searching "J.J. Barea Height" for the past 15 minutes.

So I decided to find a photo and settle the issue once and for all.

I then used the basketball as my metric. Wikipedia states that an NBA basketball is 29.5 inches in circumfe…

### Finding Blissful Clarity by Tuning Out

It's been a minute since I've posted here. My last post was back in April, so it has actually been something like 193,000 minutes, but I like how the kids say "it's been a minute," so I'll stick with that.
As I've said before, I use this space to work out the truths in my life. Writing is a valuable way of taking the non-linear jumble of thoughts in my head and linearizing them by putting them down on the page. In short, writing helps me figure things out. However, logical thinking is not the only way of knowing the world. Another way is to recognize, listen to, and trust one's emotions. Yes, emotions are important for figuring things out.
Back in April, when I last posted here, my emotions were largely characterized by fear, sadness, anger, frustration, confusion and despair. I say largely, because this is what I was feeling on large scales; the world outside of my immediate influence. On smaller scales, where my wife, children and friends reside, I…

### The Force is strong with this one...

Last night we were reviewing multiplication tables with Owen. The family fired off doublets of numbers and Owen confidently multiplied away. In the middle of the review Owen stopped and said, "I noticed something. 2 times 2 is 4. If you subtract 1 it's 3. That's equal to taking 2 and adding 1, and then taking 2 and subtracting 1, and multiplying. So 1 times 3 is 2 times 2 minus 1."

I have to admit, that I didn't quite get it at first. I asked him to repeat with another number and he did with six: "6 times 6 is 36. 36 minus 1 is 35. That's the same as 6-1 times 6+1, which is 35."

Ummmmm....wait. Huh? Lemme see...oh. OH! WOW! Owen figured out

x^2 - 1 = (x - 1) (x +1)

So $6 \times 8 = 7 \times 7 - 1 = (7-1) (7+1) = 48$. That's actually pretty handy!

You can see it in the image above. Look at the elements perpendicular to the diagonal. There's 48 bracketing 49, 35 bracketing 36, etc... After a bit more thought we…