The Big Sky Theory

mathOn any given day, there are about 87,000 flights undertaken, and at any single moment, there are between 5 and 10 thousand airplanes (commercial and private) in the skies over the United States alone. According to the FAA, on an average day, controllers handle 28,537 commercial flights, 27,178 private flights, 24,548 “for hire” flights, 5,260 military flights, and 2,148 cargo flights.   And these numbers don’t include private pilots who choose not to talk to ATC, as I often do when out cruising the neighborhood or when flying around non-towered airports.

There’s so many airplanes up there at once it’s a wonder they don’t bump into each other more often.   They don’t, it seems, because relative to the sheer volume of atmosphere in which they fly, all those airplanes actually don’t take up a lot of space.   The relative volume of airplane to the volume of sky in which they fly being the reason that they don’t bump into each other more often is called the “big sky theory.”     And statistically, given the ratio, the chances of one airplane bumping into another should be close to zero.

But although it is happening less and less, it does happen, roughly a dozen times a year, especially in crowded airspace (such as busy airports) where airplanes are more likely to converge. The big sky theory, it appears, doesn’t work that well, because the statistical probability of it ever happening is very close to zero.

Once, at a party in the living room of the Victorian house I was renting as a student with several roommates in Santa Cruz, California the math instructor and brilliant folk music satirist Tom Lehrer entertained us by demonstrating statistically that it was impossible to get wet when walking through the rain.   Perhaps it was the blackberry brandy that mysteriously found its way from a bottle in my back pocket to my tummy that prohibited me from understanding the arithmetic, but his statistics appeared impeccable and his argument was compelling.

Now, I may not be able to tell you the formula for chi-square off the top of my head, but I can work my way around ANOVAs, MANOVAs, and one of my favorite statistics (and Russian movie stars)—ANACOVAs, with fluency. Compared to highly trained academic statisticians, I still sit at the kid’s table, but I retain some perhaps egoistic pride in my ability to do discriminant function analyses, and I can work my way around most research articles I read.

The big sky theory doesn’t work for similar reasons that you really can’t wet when walking through the rain.   It is very easy to misunderstand (to be generous) or deceive (to be cynical) with statistics.   (I am fond of “proving” to kids that I have 11 fingers by counting down from 10 on one hand and then adding five when I get to the other.)

That is why Joel Best’s book “Damned Lies and Statistics” and its subsequent editions should be required reading for anyone who reads anything, pretends to know something, and hasn’t studied statistics. It should also be required reading for journalists, with whom I have particular antipathy for perpetrating the most heinous of statistical misstatements.

Theories can be extremely convincing, especially when backed by statistics.   As an autism “expert,” I once described in detail the theory behind how the preservative thimerosal, used in the MMR vaccine, can cause autism.   I had a room full of family practice residents convinced, possibly because I sprinkled the explanation with statistics. (The proportion of thimerosal in vaccines, the multiples of mercury based on the FDA’s own limits of safety, the correlation between mercury poisoning and autism symptoms, etc.)   The theory can be made to look rather compelling, but it’s just wrong. These residents were smart cookies, but I could have just as easily convinced them that I had 11 fingers.

One of the many problems with statistics is that it is a very poor method for predicting low-frequency events, such as rain in California, earthquakes, violent behavior, or midair collisions. It is nearly impossible to account for all the variables required for a low-frequency (or extremely complex) event to occur.

The driveway to my domicile is located a half-mile up from a highway.   Although I typically drive that half-mile slowly, the other day I had to swerve to avoid a squirrel that decided to dart in front of my car.   Sadly for both me and the squirrel (but mostly the squirrel), we collided. If I had to create a statistical model that would attempt to predict the likelihood of me colliding with a squirrel down that half-mile stretch of road, I can assure you that it would reveal that colliding with a squirrel could not happen in thousands of lifetimes.   Statistics, it seems, cannot take into consideration the notion that squirrels appear to have a robust death wish, or have a secret ritual in which the transition to adult squirrelhood is marked by darting across a road in front of Lexus crossovers with balding drivers.

So, you see, it isn’t that difficult to prove, statistically, that it is nearly impossible to get wet when walking in the rain.   And really, it should never be necessary to look out your window when piloting an aircraft because the chances of bumping into another airplane are infinitesimal.   If you believe the statistics, that is.