Chad laments that we don’t hear that much about the decathlon any more, because Americans aren’t really competitive. I also think it’s a shame, because any sport in which your score can be a complex number deserves more attention.
Yes, it’s true. The decathlon combines ten different track and field events, so to come up with a final score we need some way to tally up all of the individual scores so that each event is of approximately equal importance. You know what that means: an equation. Let’s imagine that you finish the 100 meter dash in 9.9 seconds. Then your score in that event, call it x, is x = 9.9. This corresponds to a number of points, calculated according to the following formulas:
points = α(x0–x)β for track events,
points = α(x–x0)β for field events.
That’s right — power laws! With rather finely-tuned coefficients, although it’s unclear whether they occur naturally in any compactification of string theory. The values of the parameters α, x0 and β are different for each of the ten events, as this helpful table lifted from Wikipedia shows:
Event α x0 β Units 100 m 25.437 18 1.81 seconds Long Jump 0.14354 220 1.4 centimeters Shot Put 51.39 1.5 1.05 meters High Jump 0.8465 75 1.42 centimeters 400 m 1.53775 82 1.81 seconds 110 m Hurdles 5.74352 28.5 1.92 seconds Discus Throw 12.91 4 1.1 meters Pole Vault 0.2797 100 1.35 centimeters Javelin Throw 10.14 7 1.08 meters 1500 m 0.03768 480 1.85 seconds
The goal, of course, is to get the most points. Note that for track events, your goal is to get a low score x (running fast), so the formula involves (x0–x); in field events you want a high score (throwing far), so the formula is reversed, (x–x0). Don’t ask me how they came up with those exponents β.
You might think the mathematics consultants at the International Olympic Committee could tidy things up by just using an absolute value, |x–x0|β. But those athletes are no dummies. If you did that, you could start getting great scores by doing really badly! Running the 100 meter dash in 100 seconds would give you 74,000 points, which is kind of unfair. (The world record is 8847.)
However, there remains a lurking danger. What if I did run a 100-second 100 meter dash? Under the current system, my score would be an imaginary number! 61237.4 – 41616.9i, to be precise. I could then argue with perfect justification that the magnitude of my score, |61237.4 – 41616.9i |, is 74,000, and I should win. Even if we just took the real part, I come out ahead. And if those arguments didn’t fly, I could fall back on the perfectly true claim that the complex plane is not uniquely ordered, and I at least deserve a tie.
Don’t be surprised if you see this strategy deployed, if not now, then certainly in 2012.
Maybe the solution would be to hand imaginary medals to the athletes with the greatest imaginary comaponents in their total scores.
OMG…. didn’t know about this system! Great post.
lol, gem of a post
This post rules.
I’ve said many times to anyone who will listen: when you are dealing with stuff where negative numbers don’t make sense, take a log. In the case of the decathalon, we would be talking about a weighted geometrical average. Makes far more sense.
Thus, we get rid of alpha and x0, just keep the beta terms and *multply* the results together rather than adding them. For timed events, the betas would be negative, of course.
@ jmchez (post #2)
“Wouldn’t it just be easier to compare the athlete’s performance as a fraction of the world record for that event?”
In addition to the comments presented earlier in this thread (the point system should be non-linear because it becomes harder and harder to improve your result when you get better and better), there is also the almost trivial fact that world records tend to evolve in time. What happens if the world record of 100m sprint improves while the decathlon competition is still going on? Will the points from the 100m run in the decathlon race change accordingly? What happens to the old decathlon results when the world records improve later on? How can you meaningfully compare results from different decathlon competitions (necessary if one wants to talk about the world record of decathlon) if the way athletes are rewarded points has changed?
In my opinion, those issues make it impossible to base the decathlon point system to the world records of the individual events.
Oh, and I believe the world record quoted by Sean is off, the 9000 point mark was achieved already quite a few years ago.
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I score this post at 1337. That was fantastic. Thank you for making me laugh. I have always thought that any event that needs a judge shouldn’t be an event, but it looks like every event needs a mathematical jury now.
In the currently highly competitive field of Olympics posts, this one wins. Fabulous (and educational). Thanks!
[I wonder whether there’s some math basis for picking ten events that would make the ‘best’ decathlon, and what the criteris for such would be? Hmm.]
What do you mean that there aren’t any Americans who are competitive in the decathlon? Brian Clay is the reigning World Champion (and if you know anything about track and field, you know that the athletes respect World Champions as much as Olympic Champions). Here’s his resume:
2005 World Outdoor champion; 2008 World Indoor champion; 2008 Olympic Trials champion; 3-time USA Outdoor champion (’04, ’05, ‘08); 2004 Olympic silver medalist; Two-time World Indoor silver medalist (’04, ’06); 2000 NAIA champion; two-time NAIA LJ champion … not to mention that Chris Huffins (recently retired) and Tom Pappas (2003 World Champion) ….
#13 I did not get your joke – However I think you are correct in that there should be a score for “trying your best”
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Thank you, but I think I’ll stick with theBucky Katt point system.
The ranking system defies logic. A real super analysis on a real flop metric!
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Hello… I tried to manually pingback to this entry, but it did not work, so I’ll have to go with the “poor man’s trackback” (need to do this at several bloggers’ entries – among other to-dos – being stuck here in St. Paul, too drained to blog, this might be a good time for that).
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