## Power of running world records

Following a few entries on sports here and there, I was wondering what kind of law follow the running records with respect to the distance. The data are available on Wikipedia, or here for a tidied version. It collects 18 distances, from 100 meters to 100 kilometers. A log-log scale is in order:

It is nice to find a clear power law: the relation between the logarithms of *time T* and of *distance D *is linear. Its slope (given by the *lm* function) defines the power in the following relation:

Another type of race consists in running backwards (or *retrorunning*). The linear link is similar

with a slightly larger power

So it gets harder to run longer distances backwards than forwards…

It would be interesting to compare the powers for other sports like swimming and cycling.

Jérôme Lêsaid, on 9 August 2011 at 10:37Nice and impressionnant!

Brendansaid, on 9 August 2011 at 14:37There’s a nice paper on power laws like this listed below. It appears that race time predictors like McMillan’s one use a power law similar to the one that you found.

“Approximate Law of Fatigue in the Speeds of Racing Animals” A. E. Kennelly

Proceedings of the American Academy of Arts and Sciences, Vol. 42, No. 15 (Dec., 1906), pp. 275-331.

Julyan Arbelsaid, on 10 August 2011 at 10:07Thanks for the reference. This is indeed a well-known relation, as studied in this paper:

“Power laws and athletic performance”, JS Katz and L Katz

Journal of sports sciences, 1999, 17, 467-476

The authors analyzed the evolution of c and n in T=cD^n along time (and show that the power n is decreasing with epoch)

Brendansaid, on 10 August 2011 at 17:25Thanks for that reference. The paper looks very interesting.

The power law of world records « Epanechnikov's Blogsaid, on 9 August 2011 at 15:21[…] here (statisfaction […]

johnsaid, on 9 August 2011 at 20:07http://cscs.umich.edu/~crshalizi/weblog/491.html

In particuar: “Lots of distributions give you straight-ish lines on a log-log plot. “

Julyan Arbelsaid, on 10 August 2011 at 10:08Thanks for your very interesting paper!

Kensaid, on 31 August 2011 at 22:12Savaglio, S. and V. Carbone (2000). “Human performance: Scaling in athletic world records.” Nature 404(6775): 244-244.