Professional basketball players tend to be tall - very tall. The average NBA player's height is 201cm, or 6'7". In contrast, professional rock climbers in the International Federation of Sport Climbing (IFSC) have very ordinary heights compared with the general population:

I wanted to get a rough feel for how much of a role height plays in these sports, so I made a simple model to describe it. The model could be applied to any measurable, immutable talent - not just height.

I also went through the trouble of scraping the professional athlete's heights that you see in the above histograms. I obtained a list of climbing athletes from the top bouldering and lead competitors for 2018 on the IFSC site. For basketball athletes, I used the full roster for NBA and WNBA during the 2018-2019 season. I obtained heights either from the respective association's website if available, or from Wikipedia.

The model I made describes the probability distribution of heights, given two factors: the sport's selectivity and the proportion of athletic ability that is due to height. If a prior on these factors is chosen, the model can also take height data to give tighter credible intervals for their values.

My results:

It is unclear whether height helps or hurts professional climbers, but almost certainly less than 1.4% of performance is due to a height advantage either way (for both men and women).

Height helps NBA players, accounting for 27.9 to 35.0% of their performance.

Height helps WNBA players, accounting for 35.5 to 43.4% of their performance.

I surprised me a little that the distribution of climbers' heights was so perfectly ordinary (my initial hunch was that they tended to be a little on the shorter side). I know less about basketball, but found it interesting that height plays a significantly larger role in WNBA than NBA. I discuss this more below.

Modeling Natural Talent

To keep this model as simple as possible, let's make some assumptions:

An athlete's quantified ability is the sum of two values: the influence of their height, and the influence of all other independent factors.

The influence of height is proportional to height.

Height is normally distributed. The influence of all other factors is normally distributed.

Professional athletes have the highest ability of anyone who would be professional if they could.

You might be skeptical at this point - in sports where height is a strict advantage, wouldn't even the tallest people still need to be particularly talented and hard-working to become professionals? Apparently not, since roughly 17% of men over 7 feet tall (213cm) aged 20 to 40 are in the NBA.

Using our assumptions, an athlete's ability is

for someand, whereis height andis the influence of other factors. We have also assumed thatTo simplify this, we can shift and rescaleintosuch thatwhere bothandare standard normal random variables.

I devised this model so that is the proportion of ability explained by height, and the sign of indicates whether height helps or hurts.

By our assumption, professional athletes have in the top proportion of the pool of athletes. Let's call the sport's selectivity.

I have some rough estimates for it:

for the NBA, since there are about 200 players and probably about 200 million men who have tried basketball and would play professionally if they could.

for the NBA, since there are about 200 players and probably about 200 million men who have tried basketball and would play professionally if they could. for competitive rock climbing, since there are about 30 professional competitive athletes in either gender (people who are professional mainly for their non-competitive achievements don't count) out of a pool of about 3 million people who have tried climbing and would be professional if they could.

Since both and are normally distributed, ability is distributed like a random normal with variance . This means the cutoff to being in the top of potential athletes is having a of at least where is the normal cumulative distribution function. By Bayes' theorem, the probability distribution for normalized height , given that someone is a professional athlete, is

Implications

By definition,is simply the unit Normal distribution function. Andis the probability that all other aspects of the athlete's ability exceed, or the probability that a random unit Normal exceeds

This model isn't perfect, but it gives a simple, interpretable, and plausible estimate of how much a professional athlete (or any sort of outlier) owes to one of their characteristics. You can play around with the values of and below. Try to choose them such that the distribution resembles one of the histograms of athletes' heights above.