Reader Paul Mullen writes:
I was reading the WikiPedia page on “accurizing” firearms, and came across a claim I find suspicious:
Statistical likelihood says the fewer shots that are fired, the smaller the dispersion will be. 3 or 5-shot groups are acceptable for zeroing the sights and rough accuracy estimates, but most shooters consider 10-shot groups to be the minimum for accuracy comparisons.
For standardization, it is best to fire five-shot groups with the same aiming point. It is a statistical fact that group size will increase with the number of shots fired.
If I’ve learned anything from reading wmbriggs.com (and I do try), it is that probability distributions quantify our uncertainty about a future event, rather than describing how things would look if we could observe an infinite number of those events. In other words, we expect there’s a better chance that the next shot will be close to the last, rather than far (relatively speaking). But there’s still a chance that the second shot fired could be an outlier, while the 48 shots after it could all land on the same hole. Is there any reason to believe that the more shots we fire, the better idea we’ll have of the “true” group size?
First, I think we can all agree that accurizing is one of the worst gerunds since concretizing: it should be shot and put out of its misery.
Second, group size. As defined by Sniper Country: “It is the maximum distance between the centres of the two farthest shots in a group. The easiest way to do this is to measure from the outside edge of one bullet hole to the inside edge of the farthest one away.”
The claim “It is a statistical fact that group size will increase with the number of shots fired” is true. Fire one shot. What’s the group size? Zero inches. Fire a second shot. Unless you Robin Hood it (and there is a chance this happens) the group size for both shots will be larger than the group size for just one shot. The more shots you fire the greater the chance the rounds wander from their appointed path, and thus the larger the group size.
Probability does, as you say, quantify our uncertainty about future (or really any) events, even in cases where we have less than infinite information—which we never do. And experience does show that successive shots will be closer to one another than shots taken at widely different times.
It’s your last question that is of most interest: “Is there any reason to believe that the more shots we fire, the better idea we’ll have of the ‘true’ group size?” There is no such thing as true group size, but the more shots fired under a set of conditions the better we can characterize our uncertainty in what the group size will be.
There are sets of conditions; premises, if you like (to fit in with our usual language) which describe the “state of firing”. These includes physical conditions like (of course) weapon type, ammunition, position, distance to target, even the weather. But it also includes many intangibles such as your mood, blood alcohol content, health, and your interest in hitting the target, which might include noting whether the target is firing back at you, and so on.
Take a set of conditions and a number of rounds fired and marry that to some historical performance measurements and we could in theory calculate the probability (maybe only crudely) that the group size will equal some specific number, or that it will be between two given numbers. The number of rounds fired is crucial, because as we’ve already seen, the more shots the larger the group size usually.
This is different than actually measuring group size under a set of pre-specified physical conditions, as in competitions. There, the average of some number groups is taken, say 10 rounds of 5 shots, maybe varying the physical conditions slightly between rounds. The smallest average wins. Whatever your average is can be used as information to calculate the chance of future group sizes measurements.
Here’s some fun homework. The media, God bless them, often make much of the number of shots fired by policemen when they’re trying to take down a bad guy. Suppose the physical conditions include a bad guy firing back at the policeman. Further suppose that we know for a fact that each round the policeman fires has a 0.3 chance of hitting its target and incapacitating the bad guy. How many shots does the policeman have to fire so that there is at least a 99% chance of bringing down his target? The answer will surprise you and (probably not) delight your media friends.
Hint: if he fires only once, he has a 30% chance of bringing down his target. He does not have a 60% chance if he fires twice.