by Bobby Samuels

Playing golf in Utah is quite the ego boost. A 270-yard drive suddenly becomes 300 yards. That 150-yard approach shot? All you need now is a pitching wedge. Of course, the explanation behind this phenomenon is the change in altitude. Playing nearly a mile above sea level dramatically decreases the air pressure. With less air pushing down on the ball, the same amount of initial force propels the ball a further distance.

Baseball yields the same result. A 400-foot dinger easily clears the fence at most parks (at Fenway, it’s just 302 feet to right field). But at Coors Field? That’s a routine fly-out to center. Even with the center field wall 420 feet from home plate, Coors remains one of the most hitter-friendly parks in baseball.

The effects of altitude shouldn’t be confined to just baseball and golf, though. Theoretically, the reduced air pressure should affect all sports where balls fly through the air, such as basketball.

Basketball is a little trickier, though. Unlike in golf and baseball, where the projectiles fly for hundreds of feet, basketballs don’t travel nearly as far, meaning the effects of altitude shouldn’t be as pronounced. But, perhaps there is a countervailing force. Basketball requires more running, and because the blood retains less oxygen at higher altitudes, altitude’s effects on individual performance should perhaps be greater at these higher altitudes.

But, we wanted to look more at the ball-through-air effect and less at the running one. Effectively, rather than analyzing performance as a whole, we wanted to examine more closely just how altitude affects shooting. To do so, we chose to consider how free throw percentages changes at venues of lower elevation versus those at higher elevation (namely, Salt Lake City and Denver). The logic behind using this statistic is simple: free throws don’t involve the other team and so say much more about individual shooting than, say, field goal percentage. What’s more, free throws allow players to catch their breath, perhaps neutralizing the elevation’s effect on form or technique.

So, we looked at free throws from the 2006-2007 to the 2009-2010 seasons, which includes hundreds of thousands of shots, comparing free throw percentages at higher elevations versus normal ones. To go about our analysis, we used the null hypothesis that

FT%_{higher elevation}=FT%_{regular elevation} and the alternative hypothesis that FT%_{higher elevation}<FT%_{regular elevation}. We would expect the free throw percentage at higher elevations to be less, as the ball will travel slightly differently, and players who have grooved their stroke for lower elevations will be thrown off.

First, let’s look at Salt Lake City, which has an elevation of about 4200 feet. The overall free throw percentage in Utah was 75.5%; those who shot at least one free throw in EnergySolutions Arena made 75.8% in the other 29 venues. But, this difference wasn’t significant, as the p-value for our hypothesis test was .26. This means even if the true free-throw percentage was the same, we would expect a result this extreme 26% of the time. Of course, a more accurate indication of the effects of altitude would be the difference in free throw percentage of visiting players to Utah. This yielded a similar result: visitors actually shot slightly worse at Utah (75.9%) than away (76.1%). Again, this result wasn’t statistically significant.

Now, we’ll look at Denver. We would expect the effects of altitude to be larger in Denver than in Salt Lake City as Denver is over 1000 feet higher, sitting at an altitude of just about a mile high. But, we should expect the effects of higher elevation to be more than 25% greater at Denver than in Utah, as the air pressure varies non-linearly with altitude. Interestingly, the data confirmed this notion, as players had a much lower free throw percentage at the Pepsi Center (74.9%) than they did elsewhere (75.8%). Our hypothesis test yielded a p-value of the 0.0248, demonstrating significance at the .05 level and allowing us to reject the null hypothesis. When we compared how visiting players performed away from and at Denver, the difference became even larger. Outside of the Pepsi Center, these players shot 76.0%, but at this higher elevation, they shot just 74.5%. Once again, our results were significant, yielding a p-value of 0.0143.

We also looked at how elite players brave the altitude. Specifically, we analyzed how the best 50 active free-throw shooters fared at Denver and in Salt Lake City. Perhaps, we reasoned, these players have their strokes so grooved at regular elevations that the higher altitude would throw them off slightly more than their counterparts. However, we actually found the opposite. As visitors, they shot .6% and .3% better at Denver and Utah, respectively, than they did at lower elevations. However, neither of these differences was statistically significant.

Although we had enough statistical evidence to reject our null hypothesis for Denver, it should be noted that this is not enough to conclusively claim altitude is the cause. There could be other factors, such as the distribution of players who got foul shots at the arenas, that is causing this difference. Still, the fact that nearly every comparison between arenas found lower free-throw percentages in Salt Lake City and Denver indicates that it is very possible that altitude is altering shots.

Did you look at individual free throw percentages for other arenas? I would think it would be important to know what the variance is between arenas – either from true statistical variance (which can be estimated, I guess) or from other factors such as different backdrops, crowds, etc.

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I’m not sure the theory that altitude would cause the ball to fly differently from that short of a distance can be proven by looking solely at free throw %. You’d have to eliminate another possible cause and that’s fatigue in response to lower levels of oxygen at higher elevations.

By studying another stat, eg blocked shots, which would be impacted by fatigue but not the flight of the ball, you could narrow down potential causes for any differences in FT %.

Also, we are producing a basketball with embedded sensors that actually can tell you the impact of altitude on the flight of a ball from any distance (note: we have not run this experiment).

I’m guessing altitude-related fatigue would be the driver here… It’s harder to catch your breath in Denver. It’d be interesting to see how FT% changes throughout the game, both in Denver and elsewhere.

Air density, which is what matters for drag effects, varies linearly with altitude, at least between sea level and Denver. So if not effect is seen in SLC, I doubt one would see an effect in Denver. Although I have not done the calculation (it would be very easy to do so), I have a hard time believing that differences in aerodynamic effects (mainly drag) play any role whatsoever in the difference in FT%.

Fatigue as the major contributor also pans out by the fact that the visitors tended to do worse than Denver at the Pepsi center. The Nuggets for the most part likely live and train in Denver, and over the course of the preseason and season their bodies adjust to playing at that higher altitude (more hemoglobin in the blood, etc). The visitors don’t have that luxury, and as MattieShoes said, they get winded…