## Thursday, April 08, 2010

### Baseball violates the rules of mathematics!!

(Looking for a roomate for STOC. Check out this site..)

Baseball Season started this week. I want to point out that Baseball violates mathematics in two ways.

1) By the rules of the game Home Plate is a right triangle with a square adjacent to it. And what are the dimensions of this right triangle? They are 12-12-17. BUT THERE CANNOT BE A 12-12-17 RIGHT TRIANGLE!!!!

2) (Information in this point is from Bill James Article The Targeting Phenomenon.) A players batting average is what percent of the time he or she gets a hit (its a bit more complicated since some things don't count as at-bats: walks, sacrifices, hit-by-ball, maybe others). You might think that the higher the number the less players achieve that batting average. Let N(a) be the Number of players with batting average a over all of baseball history. You might think
N(296) ≥ N(297) ≥ N(298) ≥ N(299) ≥ N(300)
But you would be wrong.
1. N(296)=123
2. N(297)=139
3. N(298)=128
4. N(299)=107
5. N(300)=195
There so many more players batting 300 then 299!. There so many more players batting 300 then 298!. There so many more players batting 300 then 297!. There so many more players batting 300 then 296!.

This would seem to violate the very laws of mathematics! Or of baseball! Or of baseball mathematics! Actually there is an explanation. Batting 300 has become a standard that players try to achieve. If you are batting 300 and it is the last week of the season you may become very selective on what balls you hit, you may ask to sit out a game, you will do whatever you can to maintain that 300. Similarly, if you are batting 296-299 then you will do whatever it takes to get up to 300.

This happens with number-of-hits (with 200 as the magic number), Runs-batted-in (with 80,90, and 100 as magic numbers), for pitchers number-of-strikeouts (with 200 and 300 as magic numbers), and wins (with 20 as the magic number).

If we all had 6 fingers on our hands instead of 5 then there would be different magic numbers.

So what to do with this information? Model it and get a paper out. Hope to see it at next years STOC.

1. I doubt such a paper would get accepted at STOC.

Maybe at "Innovations in Computer science" ;-)

[Just making a cheap joke, I actually like the idea of ICS]

2. The Anonymous One10:37 AM, April 08, 2010

If not STOC then at least on Arxiv or ECCC. It would probably be widely read and discussed.

3. Is this in lifetime average or yearly average? Particularly in the latter case (and even anyways, given bench players/short careers) I'd expect this to be partially explainable by the fact that if you take two small random numbers, the ratio is more likely to be .3 then .299. You can get a .300 average in 10 at bats, you need 1000 (ignoring rounding for now) to get a .299.

4. Paul, Bill James claims that he doesn't see the same effect at .286, which would be 2/7. (You can read the article in Google Books.) I'm not sure I buy it, though, because I don't have the actual data.

5. within a reasonable realm of numbers, 12, 12, 17 is an integral triangle closest to a half square. you know \sqrt(2) is not integral. so there would be some approximation in describing it in the common language, unless you require a math exam to understand the rules of baseball.

6. 1) Its yearly averages.

2) I think the article did mention
that 300 might be a more common average
than 299, but that the difference is
SO huge that this would not account for it.

(I assumed it wasn't online since it wasn't at Bill James Site which does have other things online.)

7. Another place this phenomenon occurs: U.S. congressional (and maybe Presidential) elections. If, say, there is a Republican incumbent who won with (say) 58% of the vote in the last election, and the "national swing" looks to be around 4-5% in the Democrats' favor, you're far more likely to see a swing of close to 8% in that particular district, since it's a vulnerable seat and the national party will put extra resources into trying to win it.

8. Caveat: I don't have the same solid mathematical evidence as Bill James, but eyeballing my data this appears to be the case. I still don't know of a robust way to model it however.

9. Maybe the rules of baseball take into account the curvature of the earth-- it it's curved enough there could be such a triangle

10. Could the phenomenon about batting averages have anything in common with Benford's law?

11. How bad is the 12-12-17 triangle? (Note that 12²+12²=288, and 289=17².)

Well, one right triangle is 12-12-16.97 (where 16.97 stands for the exact value 12√2). Another right triangle is 12.02-12.02-17 (where 12.02 stands for 17/√2). And the angle in the 12-12-17 triangle is
cos⁻¹(-1/288), which is 90.2°.

All considered, not too bad, I think. (In fact 17/12 is one of the continued-fraction convergents of √2, which are 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408…)

12. You don't need to go to 1000 to get a .299 average. 29 out of 97 will work.

13. Now that LANCE is back, is there a way to discourage GASARCH from posting so often?

14. Another one that may violate common mathematical definitions: Slugging Percentage. A perfect Slugging Percentage is 4.000, if it were a *true* percentage, then wouldn't it be normalized to [0,1]?