The Power Law in Sports
The Power Law shows up all around us in nature and physics, but what about sports?
The 2024 women’s NCAA basketball March Madness finals drew a record-shattering 18.9 Million TV viewers, propelled by Caitlin Clark above the viewership for the men’s NCAA final and any NBA game since 2017(!). About a month later, Leo Messi came to play at Gillette Stadium (against my theoretically beloved New England Revolution), a match that achieved a record 65,000 in attendance. The NCAA women’s final viewership is particularly incredible, and after my mind was done boggling at Clark’s individual star power (ha) having an outsized impact on the ratings of her games, it got me wondering about other instances throughout sports where these types of exponential statistical explosions occur.
The Power Law is a statistical distribution that most people are probably visually familiar with, where one value varies as a power of the change in another variable (i.e. exponentially). A linear change in one variable leads to a greater-than-linear change in a dependent variable, and when the dependent variable’s exponent is negative, you get a stereotypical Power Law distribution as seen throughout this article. The Power Law shows up all around us in physics, and in nature— from earthquake strength to human height and animal lifespan distributions— and it turns out that power distributions are pervasive in sports as well.
A great example sits at the top of this article, where Caitlin Clark ended her college career with about 500 more points than her next closest all-time women’s scoring record contenders. She was roughly 1.12x (12%) more prolific at scoring than anyone else in history, and (throwing an example dart here) might also be 1.5x (50%) more well-known than any other women’s NCAA basketball player (again, the actual number is likely lower). However, this popularity culminated in a viewership draw that was nearly 2x that of any other NCAAW final in March Madness history— and what’s more, when ranked by order of viewership, most NCAAW finals over time leveled out in a similar range around 3 million viewers, forming a stereotypical long tail:
When data are organized in descending order, the result is the Power Law distribution that you see above, where one of your variables (in this case Total Viewers) exponentially ramps at the extreme of your underlying variables, like superstar popularity or scoring proficiency. For this chart, and all those following in the article, I’ve removed x-axis data labels to be able to focus on the distribution itself— especially when looking at 100+ data points, it becomes a bit of a mess…
This type of distribution shows up all over, and in an effort to limit my writing for the week to accommodate some travel (and to show off some fun visuals, of course), I wanted to share some of my favorite Power Law distributions I’ve hunted down throughout the world of sports performance and entertainment. This search also revealed a few linear distributions, and despite my best efforts, I had to write down some thoughts about why those exist as well. This canvassing of statistical distributions throughout the world of sports deserves a much more comprehensive effort, and I may revisit this topic in future articles.
But anyways, without further ado, here are some other examples of the Power Law rearing its non-linear head throughout the world of sports:
NBA All-time Scoring Leaders
There’s no doubt that dominant athletes in the NBA have much higher output than their peers in terms of points. What surprised me though, was how dramatic the ramp is. The player with the 50th most points of all time (Tom Chambers) has just 1/2 the points of the leader (LeBron James). To get to 1/2 of Chambers’ point total, you end up going all the way to the 398th best scorer ever— quite a slow descent after a steep initial drop.
NHL Online Fandom
In one of the funnier examples I found, NHL teams’s Facebook followers demonstrates a pretty stereotypical power curve. The Chicago Blackhawks top the charts with ~2.5 million Facebook fans, and the Seattle Kraken came in last place, with just 162,000 fans. Why is this curve non-linear? Who knows.
Olympic Medals by Nation
Olympic medal distribution by country is possibly the most extreme example of the power law in sports, especially in the summer Olympics where the US dominates the total medal count, to a level that is practically un-fun:
Marathon World Record Pace
The Power Law also shows up in the progression of the world record for most distances across most sports. A steep descent in the record for a race distance eventually flattens out as more and more athletes compete with better training and technology. You eventually enter a regime of marginal gains, where even once-in-a-generation talent is only just barely better than their competitors— the dreaded zone of “marginal gains.”
And, of course, some more linear statistical distributions continue to anchor themselves the world of sports as well, just hoping to one day become a power distribution:
Ultrarunning Course Records
As opposed to the traditional marathon distance which has been run by tens of thousands of elite athletes for over 100 years, most ultramarathons have only been held for a few dozen years, with elite athletes only attending a few events each year due to their required training and stress on the body. This means that athletes are still realizing major gains in the course records for most distances, in a relatively linear fashion, as shown by the Western States women’s course record over time (apart from the inaugural race).
I expect this linear trend for most ultramarathons to begin flattening in a manner similar to traditional marathons as more and more athletes compete under ideal conditions over time.
MLB Local Ratings
As opposed to Facebook fandom for NHL teams, which follows a near-perfect power distribution, local TV ratings for MLB teams are distributed almost perfectly linear. Similar to the NHL fandom example, I’m not entirely sure why this is the case, but there may be a research paper floating out on JSTOR somewhere explaining the difference!
Super Bowl Victories by Team
The NFL vocally prides itself on having high levels of parity, which results in a pretty linear distribution of Super Bowl appearances across its teams. I think part of the reason is that there haven’t been that many super bowls yet— fewer than 60 total— and turnover in team talent, management, etc. means that it’s difficult to remain differentiated for decades at a time.
So apart from taking in some fun charts, what is there to actually take away from this information?
This is a question whose answer could easily destroy my goal to have a light word count this week, so I will punt some of the discussion to a future article. However, to possibly state the obvious, I think it’s critical to simply understand the distribution (power vs. linear) of your critical target metrics and quantifiable goals relative to history and the competition. That basic knowledge can often dictate your overall strategy for pursuit of improvements.
Are you at the long tail of an exponential distribution and need to examine every last drop of performance improvement to gain a 0.05% edge over the competition, or are there structural or fundamental aspects of your team composition, training, fueling, or tech that you could change to realize a 15% linear advantage? Could taking risks on generational talent over and over again be a smarter pursuit than a more balanced team-building approach for pursuit of championships and viewership? I pretty firmly believe that these types of directional questions can be answered by looking at the graphical distribution of your target outcome (points, viewership, world record, etc.) relative to your dependent variable (time, teams, count, etc.).
Another piece of the puzzle here is anticipating how and when the distribution will change in the future. It’s rare, or maybe impossible, for a Power Law distribution in sports to revert to a linear one without some major underlying rule or technology change. However, most linear distributions and progressions will eventually turn exponential over time with enough data points. That is, unless there are structural features of the competition that you can identify that hold the distribution linear!
But anyways, that’s enough fun for this week, I hope you enjoy knowing more than almost all other people about the Facebook fandoms of NHL teams!
Hey Cameron, great article, had no idea you got up to this sort of thing in your spare time! This could easily be an article in the Athletic (pre-NYT).
One minor point I might make is that Seattle has only had an NHL franchise for a couple years, could certainly help to explain their number of likes!
Cheers,
Jack (formerly with AMP)