# Exponents and the decathlon

During the Olympics, I stumbled across an application of exponents that I had not known before: scoring points in the decathlon or the heptathlon. From FiveThirtyEight.com:

Decathlon, which at the Olympics is a men’s event, is composed of 10 events: the 100 meters, long jump, shot put, high jump, 400 meters, 110-meter hurdles, discus throw, pole vault, javelin throw and 1,500 meters. Heptathlon, a women’s event at the Olympics, has seven events: the 100-meter hurdles, high jump, shot put, 200 meters, long jump, javelin throw and 800 meters…

As it stands, each event’s equation has three unique constants — $latex A$, $latex B$ and $latex C$— to go along with individual performance, $latex P$. For running events, in which competitors are aiming for lower times, this equation is: $latex A(BP)^C$, where $latex P$ is measured in seconds…

$B$ is effectively a baseline threshold at which an athlete begins scoring positive points. For performances worse than that threshold, an athlete receives zero points.

Specifically from the official rules and regulations (see pages 24 and 25), for the decathlon (where $P$ is measured in seconds):

• 100-meter run: $25.4347(18-P)^{1.81}$.
• 400-meter run: $1.53775(82-P)^{1.81}$.
• 1,500-meter run: $0.03768(480-P)^{1.85}$.
• 110-meter hurdles: $5.74352(28.5-P)^{1.92}$.

For the heptathlon:

• 200-meter run: $4.99087(42.5-P)^{1.81}$.
• 400-meter run: $1.53775(82-P)^{1.88}$.
• 1,500-meter run: $0.03768(480-P)^{1.835}$.

Continuing from FiveThirtyEight:

For field events, in which competitors are aiming for greater distances or heights, the formula is flipped in the middle: $latex A(PB)^C$, where $latex P$ is measured in meters for throwing events and centimeters for jumping and pole vault.

Specifically, for the decathlon jumping events ($P$ is measured in centimeters):

• High jump: $0.8465(P-75)^{1.42}$
• Pole vault: $0.2797(P-100)^{1.35}$
• Long jump: $0.14354(P-220)^{1.4}$

For the decathlon throwing events ($P$ is measured in meters):

• Shot put: $51.39(P-1.5)^{1.05}$.
• Discus: $12.91(P-4)^{1.1}$.
• Javelin: $10.14(P-7)^{1.08}$.

Specifically, for the heptathlon jumping events ($P$ is measured in centimeters):

• High jump: $1.84523(P-75)^{1.348}$
• Long jump: $0.188807(P-210)^{1.41}$

For the heptathlon throwing events ($P$ is measured in meters):

• Shot put: $56.0211(P-1.5)^{1.05}$.
• Javelin: $15.9803(P-3.8)^{1.04}$.

I’m sure there are good historical reasons for why these particular constants were chosen, but suffice it to say that there’s nothing “magical” about any of these numbers except for human convention. From FiveThirtyEight:

The [decathlon/heptathlon] tables [devised in 1984] used the principle that the world record performances of each event at the time should have roughly equal scores but haven’t been updated since. Because world records for different events progress at different rates, today these targets for WR performances significantly differ between events. For example, Jürgen Schult’s 1986 discus throw of 74.08 meters would today score the most decathlon points, at 1,384, while Usain Bolt’s 100-meter world record of 9.58 seconds would notch “just” 1,203 points. For women, Natalya Lisovskaya’s 22.63 shot put world record in 1987 would tally the most heptathlon points, at 1,379, while Jarmila Kratochvílová’s 1983 WR in the 800 meters still anchors the lowest WR points, at 1,224.

FiveThirtyEight concludes that, since the exponents in the running events are higher than those for the throwing/jumping events, it behooves the elite decathlete/heptathlete to focus more on the running events because the rewards for exceeding the baseline are greater in these events.

Finally, since all of the exponents are not integers, a negative base (when the athlete’s performance wasn’t very good) would actually yield a complex-valued number with a nontrivial imaginary component. Sadly, the rules of track and field don’t permit an athlete’s score to be a non-real number. However, if they did, scores might look something like this…

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