Big Men Can(’t) Shoot

It’s almost a fact amongst NBA fans — the taller you are, the worse you are at shooting. And if you look at the stats, it would indicate that this is true. If you are an NBA player, being tall is correlated with a lower FT%.

But then there are the exceptions: KD, Bird, Dirk, Peja Stojaković — all top 20 from the line all time, yet all over 6’9”. So are these guys just exceptions — or is the heuristic we all use wrong?

I think the latter is true. So lets see what would happen if these two stats were unrelated, and how this negative correlation can occur.

Skills vs Height

Let’s say that both free throw percentage and completely independent of each other, what would the distributions of these look like across 10000 people?

As expected, two very normal-looking normal distributions — bell curves. But more interestingly, this is what happens when we look at each individual’s height compared to their free throws. Obviously, no correlation.

Not Everyone Makes the NBA

The crux of this perceived relationship comes down to one fact: not everyone makes the NBA. Of the 350,000,000 people who live in the USA, there are only 450 NBA players, about 0.00001% of the population. In our simulation of one million people, this would be 100 people. So, if you were picking the best 100 people to represent this population at basketball, who would they be?

In very simple terms, it’s going to be the people who are tall, or the people who are good at free throws. Let’s say the quality of a player is the sum of their percentiles in these two stats.

It’s pretty clear here that the highest quality players are going to be the ones who are both quite tall and pretty good at free throws, but often not the tallest players or the best shooters. There are going to be plenty of tall players who just can’t shoot – the world’s tallest man never played in the NBA – while there are plenty of shorter players who are exceptional shooters but too short to ever make the league. So let’s pull out the top 100 players, who are all in the ‘NBA Zone’ on this distrubution.

Shockingly, we can see the downward correlation. Even though there is nothing relating free throw percentage and height, after we select the top 100 players, this correlation emerges. The $R^2$, 0.13, matches this other analysis, which was done on real NBA data, so I’m pretty happy to say that this is what is happening in real life. It’s also cool to see the outliers emerge, with the highest quality player being a 203 cm tall player who shoots 15% better han other people their height – a clear Dirk or Slim Reaper emerging from these two independent distributions.

For the statistic nerds, this is just an example of selection bias, and a good one to steal as an example