I would like to build upon an excellent posting somewhat well known to gun people who hang out online: https://www.ar15.com/forums/t_3_16/512887_.html&page=1
A couple points:
-- the extreme spread of a pattern of shots on target is statistically just about useless.
-- The 'mean radius' method posted at the link gets use closer to an actual valid method of determining accuracy, but it also has flaws, which will become clearer in a moment.
First, ask yourself: what is it that we actually want to determine in terms of a numerical description of "accuracy"? (note, what we mean by accuracy is technically precision, but let's not get too pedantic).
The answer is: we want to know that probability that my next shot will fall within some predictable distance of the point of aim. Want to map POI to POA in terms of numerical probabilites.
In other words, saying your rifle shoots "1 MOA" groups is itself a meaningless phrase. How many shots will fall within a 1 MOA circle centered on the POA out of 10? Or 100?
Statistically, even a terribly 'inaccurate" rifle or pistol can shoot a '1 MoA group'-- it will just take shooting a lot of groups until the stars align and the random probabilities create that 1 MOA group.
So here's better how you can determine the true 'Accuracy' (read: precision) of a given firearm and ammo combination.
First, you must define that confidence level you care about-- 90% or 95% are commonly used, as is 99%.
Then, you must shoot a statistically relevant number of shots. The minimum number actually depends on 1)what confidence level you need to have in your data and 2) how much dispersion and variation you are seeing in your groups. I'll save the math on how to determine this until some glutton actually wants to see it.
For example, if you shoot a four-round group that ends up with all holes touching at 100m, then statistically, the odds of a 5th shot being very close to the others is pretty good. However, the odds of that fifth shot perfectly touching your first shot when all the intervening shots were scattered all over? That probability is remote-- but it is STILL possible.
The problem with using a mean-- an average of anyting-- is that they are very sensitive to skew. This is why median is the preferred measure for things that a wide range of possible values, and when those values are expected to vary a great deal. Median is far more useful, as it represents the 50% confidence point-- you could expect half the values to exceed it and half to be less. That's the definition of median, actually.
An average, however, doesn't have this predictive power. If the average height of a group of people is 5'7", that tells you nothing about how many people are actually 5'7"-- just that the group would average this amount. If you have a super tall or short person, or a couple of each, the median will adjust to reflect this, while the mean does not.
So, with a proper understanding of the stats, we should be able define the repeatability of a gun/ammo combination either in terms of probability, given circle size, or -conversely-- group size, given probability.
Thus, a proper accuracy (read: precision) study would indicate the probability of a round landing instead 1" circle of POA, 2", 3" etc. OR, it could define the circle size that corresponds to 50% probability, 90%, 99%, what have you.
Here's how to do it:
-- start with finding the center of the group as in the link above
-- Measure the distance of each hit from the center. (you can use inches or cm or whatever you want, but using MOA/MIL makes the value applicable across distances from target)
-- find the average of the distances from the group center (just like in the link).
-- Then, find the standard deviation by subtracting each individual radius from the average, then average all those "deltas"-- the difference between each data point and the average.
-- now you can calculate probabilities based on that standard deviation: plus or minus 2 standard deviations from the mean is pretty much 95% probability is. 3 standard deviation is just over 97% chance. Example is here.
This method will allow you to predict with 95% confidence the size of the circle that will contain 95/100 shots.
The downfall of this method is that is assumes a perfect normal (bell curve, Gaussian) distribution. I don't know if it's valid for bullet holes to be so distributed on a target. Certainly it's very close.
If you want to go further, you can run statistical tests to find other distributions that may fit better (lognormal, Weibull, etc).
If anyone wants me to work through an example, let me know.
A couple points:
-- the extreme spread of a pattern of shots on target is statistically just about useless.
-- The 'mean radius' method posted at the link gets use closer to an actual valid method of determining accuracy, but it also has flaws, which will become clearer in a moment.
First, ask yourself: what is it that we actually want to determine in terms of a numerical description of "accuracy"? (note, what we mean by accuracy is technically precision, but let's not get too pedantic).
The answer is: we want to know that probability that my next shot will fall within some predictable distance of the point of aim. Want to map POI to POA in terms of numerical probabilites.
In other words, saying your rifle shoots "1 MOA" groups is itself a meaningless phrase. How many shots will fall within a 1 MOA circle centered on the POA out of 10? Or 100?
Statistically, even a terribly 'inaccurate" rifle or pistol can shoot a '1 MoA group'-- it will just take shooting a lot of groups until the stars align and the random probabilities create that 1 MOA group.
So here's better how you can determine the true 'Accuracy' (read: precision) of a given firearm and ammo combination.
First, you must define that confidence level you care about-- 90% or 95% are commonly used, as is 99%.
Then, you must shoot a statistically relevant number of shots. The minimum number actually depends on 1)what confidence level you need to have in your data and 2) how much dispersion and variation you are seeing in your groups. I'll save the math on how to determine this until some glutton actually wants to see it.
For example, if you shoot a four-round group that ends up with all holes touching at 100m, then statistically, the odds of a 5th shot being very close to the others is pretty good. However, the odds of that fifth shot perfectly touching your first shot when all the intervening shots were scattered all over? That probability is remote-- but it is STILL possible.
The problem with using a mean-- an average of anyting-- is that they are very sensitive to skew. This is why median is the preferred measure for things that a wide range of possible values, and when those values are expected to vary a great deal. Median is far more useful, as it represents the 50% confidence point-- you could expect half the values to exceed it and half to be less. That's the definition of median, actually.
An average, however, doesn't have this predictive power. If the average height of a group of people is 5'7", that tells you nothing about how many people are actually 5'7"-- just that the group would average this amount. If you have a super tall or short person, or a couple of each, the median will adjust to reflect this, while the mean does not.
So, with a proper understanding of the stats, we should be able define the repeatability of a gun/ammo combination either in terms of probability, given circle size, or -conversely-- group size, given probability.
Thus, a proper accuracy (read: precision) study would indicate the probability of a round landing instead 1" circle of POA, 2", 3" etc. OR, it could define the circle size that corresponds to 50% probability, 90%, 99%, what have you.
Here's how to do it:
-- start with finding the center of the group as in the link above
-- Measure the distance of each hit from the center. (you can use inches or cm or whatever you want, but using MOA/MIL makes the value applicable across distances from target)
-- find the average of the distances from the group center (just like in the link).
-- Then, find the standard deviation by subtracting each individual radius from the average, then average all those "deltas"-- the difference between each data point and the average.
-- now you can calculate probabilities based on that standard deviation: plus or minus 2 standard deviations from the mean is pretty much 95% probability is. 3 standard deviation is just over 97% chance. Example is here.
This method will allow you to predict with 95% confidence the size of the circle that will contain 95/100 shots.
The downfall of this method is that is assumes a perfect normal (bell curve, Gaussian) distribution. I don't know if it's valid for bullet holes to be so distributed on a target. Certainly it's very close.
If you want to go further, you can run statistical tests to find other distributions that may fit better (lognormal, Weibull, etc).
If anyone wants me to work through an example, let me know.