A Trip to The Gun Show; Or, An Analog Parable About Precision Versus Accuracy

Cheaper rifle

Cheaper rifle’s shots

About a year back, I went to a local gun show looking for a Springfield Model M1903 rifle. It was the model my wife’s grandfather carried in World War I, and she had asked me to find one for her uncle as he wanted one to put in their family museum. (While the idea of a family museum seems pretty strange to me, going to a gun show to procure an exhibit certainly beats painting the house.)

At the show, I found a gun dealer who had two such rifles. One cost three times as much as the other. I asked the dealer what the difference was, and he said that I could take both home for inspection if I left a deposit. I took both over to my friend Ernie’s place. Ernie, an amateur gunsmith, said he would be happy to help me with the testing. We placed each rifle in a gun vise, aimed at the target, and shot a single round (Fig. 1).

The results show that the cheaper rifle (left target) was more accurate than the more expensive one. Accuracy is defined as the degree of closeness of a measured quantity to the desired value. In this case, the measured quantity is the bullet’s position and the desired value is the bull’s eye. It surprised me that the cheaper gun would be so much more accurate. Ernie suggested that since each rifle had four remaining shells in the clip, we should fire them all. Doing so resulted in an entirely different picture (Fig. 2).

Ernie said that although it was accurate, the cheaper gun was not precise. Precision is defined as how tightly the shots are placed. Other terms for precision include repeatability, reproducibility, relative accuracy, and effective resolution. They are all different names to describe a single value: error.

The cheaper rifle was just plain noisier too. Ernie said that most likely, the cheaper gun was an actual WWI rifle that had been fired too fast in battle, causing the barrel to heat up and lose temper. With softened steel, further shooting quickly wore out the barrel. It is useless for anything other than decoration since accuracy without precision is of little use. I commented that the other rifle, although precise, was not very accurate and that didn’t seem very useful either. Ernie said that the accuracy would not be a problem

after he calibrated the sight. He made a couple of adjustments and the rifle put all five shots well within the bull’s eye. It is now both accurate and precise.

All accuracy comes from calibration, either from the calibration and strict statistical control of all the processes necessary to construct a system or the actual calibration of the system after construction.

Further Experiments
I decided I would buy the cheaper gun for my wife’s uncle since I don’t think he will actually ever fire it. But I also decided to buy the second gun for myself. I was not done with the experiments.

Continue on next page

Early the next Saturday, I went over to Ernie’s, where we quickly calibrated the rifle. We took shots every hour, and I noticed that as the temperature increased, the shots slowly moved up and to the right. Ernie said the sight had issues with temperature stability. Stability is also another term for precision, generally used to define the relationship between the error and a single parameter.

In this case, error is measured as inches per degree. (For voltage references, it is PPM/°C.) Other parameters affect precision, but they aren’t functions of the rifle. My particular rifle has adjustments for distance and windage. You can think of this as open-loop calibration.

Ernie said he would like to run another experiment. I left the gun with him, and he fired it once a month for 10 months, doing so whenever the temperature was the same. Ernie showed me that there is also stability error over time. He said the problem was with the leave spring in the sight. He offered to replace the sight with one that has better stability across both temperature and time. Of course, it will cost more. It is same with electronics. Components with better stability generally cost more.

Last week, I went to retrieve my rifle. Ernie suggested some target practice. Ernie shot the gun and kept all the shots well within the bull’s eye. I could barely keep the shots all on the target. Apparently, I add more noise to the system than Ernie does, so I’m less precise. In any system, precision depends on all its pieces.

To best define your system requirements, it is necessary to understand just what amount of accuracy and what amount of precision is needed. Other terms for precision include the effective number of bits (ENOB) and signal-to-noise ratio (SNR).

In most cases, the precision will be greater than the accuracy. Rarely is the accuracy greater than the precision. In these rare cases, oversampling can increase the precision. In many cases, using stable components with calibration will cost less than using very accurate components.