How many shots in a group?
by B.B. Pelletier
SavageSam prompted this report, but many of you have indicated an interest. The number of shots that should be in a group is determined by how confident you want to be that the group represents what the gun can do. Some people will argue that if the gun shot a particular group they don't need anything more because obviously that IS EXACTLY what that gun can do. Today, I'm going to try to explain why that is not the case.
It all comes down to statistics, specifically inferential statistics, which is the science of using a small number of data to represent a very large number. But don't worry, today's report isn't going to be about statistics.
"Here's a partial score - Cleveland 3."
Doesn't tell you very much, does it? You don't know who the other team is, or even what they're playing. If it's basketball, it's most likely very early in the game. If it's hockey, the game could be over. So, partial data isn't very helpful.
I can over-simplify this entire report in one statement. A sample size of 30 provides enough data to predict the outcome of a large number of cases to a very high degree of confidence. It doesn't matter how big your data set is; a sample size of 30 will give a very accurate picture of the entire population.
However, there are many problems--all dealing with the randomness of the sample. If, for example, you want to know the average height of adult men in the world and your sample is taken from just one tribe of Watusi, your results will be skewed toward the high end and utterly worthless. The Watusi are known to be the tallest people on earth, so the average height of 30 adult males might be 6 feet, 7 inches. That's probably taller than most of the readers of this blog. How the sample is taken plays a huge part in the reliability of the outcome.
Can we set that aside now? I just wanted to demonstrate that I know how samples can be skewed. Let's get on with the main discussion. How many shots should be in a group to give good confidence that the group represents the accuracy of a given gun?
Well, one shot is worthless for group size prediction. Do we all agree on that? One shot tells you very little about where the next shot is going. If that's true, why do many people adjust their scopes after firing a single shot? Because they do not understand how a gun works and how randomness plays into the problem.
Three shots is the next size group, and, as unbelievable as it seems - at least to me - there are gun writers getting away with publishing three-shot groups in their articles. There's even one gunmaker who provides three-shot groups to sell all his guns! They're beautiful groups, too. Very tight. When you see them you really believe the gun could shoot that well. But it can't. Not all the time, which is what a group is supposed to be telling us.
A three-shot group has just one application. When you're sighting in, shoot a three-shot group after every change to the sights. Then, adjust the sights from the center of that group. As far as I'm concerned, that's the one and only use for a three-shot group. Why? Because statistically it does a very poor job of representing how the gun shoots.
The next group size commonly used is a five-shot group. While it's just two shots more than a three-shot group, the probability that it represents how well the gun can shoot increases enormously. Is a five-shot group representative of how good the rifle can shoot? That depends on what YOU mean by how well it can shoot. I can shoot five five-shot groups that will average one size and another single 25-shot group that will be a different size. The 25-shot group is going to be larger than the average of five five-shot groups. Even though the number of shots is the same--25 in both cases--the way the groups are measured will enter into things and give two different answers. Also, because I averaged the five-shot groups, I skewed the results in a favorable direction.
So, is a five-shot group representative? Not if what you're trying to determine the gun's accuracy. If you're trying to show what size five-shot groups it can shoot, then it is representative.
Here's the deal. Five shots are only a fair predictor of accuracy. Ten shots are much better, but even then the prediction they give is not entirely reliable. Statistically, it takes 30 shots to approximate to a high degree of confidence the group size you will get if you shoot the gun 1,000 times. Twenty-five shots will group very close to the same size as 1,000 shots and ten shots will give a group that's close enough for most practical uses. Five shots will not.
With five shots, you get in the right neighborhood, but you cannot take your results to court. In real terms, that means a rifle that gives a five-shot group of one inch might give a ten-shot group of 1.4 inches and a 25-shot group of 1.54 inches. A 30-shot group from the same gun might be 1.61 inches and so might 1,000 shots. Do you see how it works? A three-shot group from the same gun might be as small as half an inch.
Who are you and what do you need to know?
There are people who absolutely must know the ultimate truth about everything, and then there are the majority of people who can tolerate a little slop for the sake of expediency. Statistics support the first group by telling them that there will be no measurable difference between 30 shots and 1,000 shots, so they don't waste a lot of ammunition trying to find out what just 30 shots will show. But even that much shooting is way too involved for anyone who publishes their results.
But sometimes there are very compelling reasons for knowing the absolute truth. Developers, for example, should want to know exactly how their product performs under normal conditions.
Five-shot groups are often used in print, realizing that they do not represent the absolute final word on a gun's accuracy, but they do put us into the right neighborhood. Just like the EPA average mileages for a given model automobile may not represent the mileage for your car, a five-shot group might only be a rough guesstimate for accuracy. If you buy a gun that shoots half-inch five-shot groups at 50 yards you should get concerned when your gun shoots only 1.5-inch groups at the same distance, just like you should get concerned if your Toyota Prius gets only 15 mpg.
When I tested my USFT rifle, the best group of JSBs at 50 yards measured 0.335" c-t-c.
The ultimate in reliable data
Ten-shot groups are used whenever the goal is to publish defensible results. And 25- to 30-shot groups are used by developers, testers and anyone who wants their results to stand up to peer review. Three-shot groups should be reserved for scope adjustment, only, and never used as a predictor of a gun's accuracy.
Test target sent with the rifle shows 25 shots at 51 yards passed through a 0.663" group.
At this point in our discussion I could show you numerous ways of manipulating the data to represent anything you want to show. But I think most of you are aware of that possibility, so I'm not going there.
You can do something about these different group sizes when you see them in print. If you see a three-shot group, double the size to get the probable 5-shot group from the same gun. To convert a five-shot group to ten shots, multiply by 1.4. To convert a ten-shot group to 30 shots multiply by 1.15.
Please don't do the math on the two groups shown above and then write me to say that what I said about the difference between five shots and 25 shots did not play out that way. You will only confuse yourself. I know it didn't work out exactly as I described. What's missing is the CONFIDENCE you can have that the five-shot group is telling you anything worthwhile.
I will continue to show five-shot groups in my reports because of time constraints, which is the big reason most writers use them.