the air up there

Back in 2004, the APS April Meeting was held in Denver, Colorado, and a group of us -- science writers, APS staffers, and a few random physicists -- took a break from the scheduled events to take in a baseball game at nearby Coors Field, home of the Colorado Rockies. There seemed to be an inordinate number of flyballs and home runs, and one of the physicists in attendance offered an interesting explanation for why this might be the case: the high elevation of the stadium. (Denver's nickname is the "Mile-High City," since it's located one mile above sea level.) In fact, the stadium has a reputation of being "home-run friendly," on a par with Chicago's legendary Wrigley field, although no one seems to attribute Wrigley's high incidence of home runs to altitude.

The physicist made a convincing off-the-cuff argument, and his outlook gibes pretty well with the accepted wisdom. Per Wikipedia, Denver's high altitude decreases the amount of air resistance on batted balls so the balls travel farther when hit. Plus, the low air pressure means the pitches "break" less severely and are therefore easier to hit. To combat this, the baseballs used in Coors Field games are actually placed in a humidor beforehand to increase their weight. (Jen-Luc Piquant offers this tidbit of random local trivia: The Rockies are known as the "Blake Street Bombers" because of the high rate of home runs -- and the stadium's location on Blake Street.)

Not being mathematically inclined myself, I was perfectly happy to take the physicist's assertions at face value. But now two mathematicians have taken a fresh look, statistically speaking, at the effects of elevation on slugging percentages in major league baseball, no doubt inspired in part by their fondness for the local stadium: Coors Field. Their analysis was published in the most recent issue of Chance, a joint publication of the American Statistical Association and Springer-Verlag. (For a 2002 article on home runs and ball parks, go here.)

Jay Schaffer and Erik L. Heiny, both of the University of Northern Colorado, based their study on a statistic known as the player's slugging percentage, or the total number of bases divided by the number of at bats, applying a statistical model to determine whether elevation was a signficant factor on the hitting statistics for both major leagues in 2003. They found that the slugging percentage in Coors Field is significantly different from that in any other ball park. In fact, it's about 9.2 percentage points higher than at middle elevations (defined as between 500 and 1100 feet), and about 12.5 percentage points higher than at elevations below 500 feet.

The Colorado mathematicians aren't the first to be sufficiently intrigued by this phenomenon to make a formal study of it. Numerous books have been published comparing hitters over different eras in baseball's history, and many of these took the impact of specific ballparks into consideration. A US Naval Academy numbers guy named Howard Penn used something called a "calculus-based gunnery model" several years ago to calculate the average initial velocity required for a baseball to clear the fence in various stadiums. He determined that because of the elevation, a baseball travels roughly 10% farther at Coors Field than it does in other stadiums. Furthermore, when Penn compared the number of home runs hit by teams playing at home versus that same team on the road, the Rockies really stood out: in 2001, 58% of their home runs occurred at Coors Field. And a full 60% of the home runs at Coors Field were made by batters from visiting teams. Over the last decade, four of the Rockies' star players have pooled their efforts to snag six national league batting titles.

Of course, there are other factors and variables that can influence slugging percentage, such as the dimensions of a given ballpark. The fences at Coor's Field are fairly distant, for instance, and in the past, analysts have reasoned that since outfielders therefore have to play deeper, the chances of hitting a single into the shallow outfield are a bit higher. Schaffer and Heiny, however, discount this factor: apparently Coors Field's dimensions aren't significantly different from those of other major stadiums, certainly not enough to account for the statistical anomaly in slugging percentages.

Other mathematicians may quibble with the methodology used by the Colorado colleagues; to some, measuring slugging percentages is a bit out of date, but most would agree that ballparks can have a significant effect on the number of runs scored, batting averages or what have you. But there will always be intangible factors that even the best statistical models can't anticipate (particularly the "human factor"). In their Chance paper, Schaffer and Heiny admit that one could add all kinds of independent values, while noting that no matter how good a particular statistical study, "Most likely the randomness of baseball never can be accounted for completely. Players go through hot and cold streaks for reasons even they do not understand."