Dear Bob,

Bruce Buck forwarded to me your message, in which you asked, “You always say that shot patterns are “Gaussian.” I know a little about statistics, and was wondering if you could provide any references explaining the math and/or experiments behind your statement.”

That’s a nice, short, intelligent question. Wish I could give you a nice, short answer, but we all know that those aren’t any fun anyway. So, as The Technoid would say, pull those boots up high?

Basically, you are asking: where is your proof that shot patterns fit the elegant, bell-shaped curve we call the Gaussian distribution? A good start would be for you to read Bruce’s definitive article, “The Technoid Takes Gauss”. As Bruce explains, the idea has been around since the 1920s, when a French artillery expert named Journee started analyzing the impact patterns of successive artillery rounds. Subsequently, Ed Lowry, John Brindle, and Roger Giblin extended the theory to shotgun patterns. Lowry published several articles some years ago in American Rifleman, and Brindle has a chapter in his book (now out of print) about it. Giblin’s work is published in several reports as well as papers in some of the British scholarly journals.

OK, where is our proof? If you were sitting here with me, I could show you several batches of data which I think would convince you. It’s harder to do via e-mail, but let me try. In my research, I started with some of my own pattern data, then went on to that of Oberfell and Thompson (O&T) in their unusual and classic book, “The Mysteries of Shotgun Patterns” (also now out of print). The strange thing about O&T is that all the pattern data in their book closely follow the Gaussian distribution, yet they never explicitly recognized it. They made no attempts to apply theory. The closest they came was to say that the data “appear to follow the laws of probability”. Loosely translated, that means the Gaussian model.

Another excellent data set comes from the pattern tests performed and published regularly by American Rifleman. With assistance from others, I’ve compiled 22 years of Rifleman data, from 1978 to 1999, consisting of over 300 high-quality data points in all. The agreement of these patterns with the Gaussian model is remarkable.

Additional support for the theory comes from statistical reasoning. The Gaussian distribution occurs extensively in nature. That’s why the most common name for it is the “normal” distribution. Based on studies of natural phenomena, theoreticians (beginning with mathematician Abraham DeMoivre in 1756, who first discovered the distribution that inexplicably was much later named for Karl F. Gauss) were able to show the following, probably the most important axiom in the entire field of statistics: data that are influenced by many small and unrelated random events are approximately normally distributed. If you are interested in the math, just pick up any statistics book and you will find it there.

This explains why the normal or Gaussian (or DeMoivrian) distribution is everywhere: stock market fluctuations, weights of 12-year old boys, daily maximum temperatures, SAT scores, and yes, shot patterns. In the latter case, the travel of each pellet is independently and randomly influenced by numerous small effects: air turbulence, aerodynamic forces on pellet flat spots, variable choke forces, and many others.

See, the long answer really is a lot more fun, right?

Best regards and happy shooting,

Warren Johnson