The Right and Wrong ways of thinking about accuracy (warning: stats)

[Hohn]

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.

 

[youngda9]

Most people just aren't going to take the effort to jump through all of those hoops. It just takes too much time.

I always think of a 1" group as only missing the point that you are aiming at by 0.5", if you have the group centered where you are aiming. I shoot several 5 shot groups, with each load that I am testing, to determine accuracy in my rifles.

 

[T.Lex]

I am interested in your beliefs. Do you have an online newsletter to which I could subscribe?

(In all seriousness, rep inbound.)

 

[Hookeye]

If I get a one inch (or half inch) rifle............that for me means the group was centered on my aiming point (or vert line, as set to hit 1" high at 100 yds).
If I set it to hit 1" high and it peels a .75 inch group 2" high (centered).............I wouldn't count that.
But then, I've never had guns do POI shifts once set.

Did have one rifle (POS Win 94 BB in .307- first yr) that thought it was a shotgun.
Another was a 742 C..............first shot 1" high at 100. Next 4 were on same vert line, but 7 or 8" low.
I'd call that a 9" group. Was repeatable from cold bore. No wonder the gun was like new.

It went bye bye rather quickly (traded to place I got it from, so as to not hose anybody).

 

[Hookeye]

I want small groups that don't shift POI. So far it's been pretty easy to get, across a few rifles.
Groundhog hunters demand it.
Not many chucks these days.
Got a new to me rifle for deer. Ruger #1 RSI (Mannlicher). How it shoots, and if weather makes for POI changes..........remains to be seen.
Def need to test for it. Reportedly can be rather touchy.
Deer gun, if it doesn't shift and holds 1.5" I'll be happy.

 

[Hohn]

Let's work through a brief statistical example of this:

Let's say you find the center of the group using the horizontal and vertical line method linked above. Then you measure each shot's distance from that group center at 100m. You shoot 10rd groups.

Here's your first group:
1) 2"
2) 1"
3) 0.75"
4) 1.5"
5) 1.25"
6) 2.5"
7) 2"
8) 2"
9) 2"
10) 1"

If you add up all those distances and calculate the average, you'd get 1.6" (16"/10). This is the "mean radius" method outlined at the link by "Molon."

But what if you had another group that also had the same "mean radius"-- is it just as accurate, or not?

Consider this second group at 100m:
1) 0.5"
2) 0.5"
3) 1"
4) 1"
5) 1"
6) 2"
7) 1"
8) 3"
9) 3"
10) 3"

Note that in this second group, the total error from group center is still 16" and the "mean radius" is still 1.6"

But which is a better group?

This is why me must use standard deviation and statistical distribution. The standard deviation of the first group is 0.5798. This is found by subtracting each value from the mean and averaging all those results (online calculators and spreadsheets make this easy)

The second group, by comparison, has a standard deviation of 1.0488. In other words, the second group-- WITH THE SAME MEAN RADIUS-- has almost twice as much variation as the first group!

Let's assume a "normal" distribution applies to the shot dispersion about the center. We apply the well-established fact that 95% of data points will lie within 2 standard deviations of the mean.

So in the case of the first group, we add two standard deviations (0.5798*2=1.1596) to the mean (1.6) and we get 2.7596 inches as a 95% confidence value.

In other words, based on the first group, we would conclude with 95% confidence that the rifle can hold a group a diameter of ~5.5"-- 95% of shots would lie within a circle of that diameter.


But for the second group (which recall, has the same mean radius), that 95% confidence circle would have a diameter of ~7.4"


Those of you with a military background may be familiar with the concept of "circular error probable" which is the same idea, just applied at 50% confidence instead of 95%. I find 95% to be more useful for my purposes, and it's something of an engineering gold standard.


So there you have it. That's why Mean Radius is not the whole story. It's not just how the *average* distance of the shots from the center of the group, but also scatter in the data points that calculate that average.

 

[bulletsmith]

Let's work through a brief statistical example of this:

Let's say you find the center of the group using the horizontal and vertical line method linked above. Then you measure each shot's distance from that group center at 100m. You shoot 10rd groups.

Here's your first group:
1) 2"
2) 1"
3) 0.75"
4) 1.5"
5) 1.25"
6) 2.5"
7) 2"
8) 2"
9) 2"
10) 1"

If you add up all those distances and calculate the average, you'd get 1.6" (16"/10). This is the "mean radius" method outlined at the link by "Molon."

But what if you had another group that also had the same "mean radius"-- is it just as accurate, or not?

Consider this second group at 100m:
1) 0.5"
2) 0.5"
3) 1"
4) 1"
5) 1"
6) 2"
7) 1"
8) 3"
9) 3"
10) 3"

Note that in this second group, the total error from group center is still 16" and the "mean radius" is still 1.6"

But which is a better group?

This is why me must use standard deviation and statistical distribution. The standard deviation of the first group is 0.5798. This is found by subtracting each value from the mean and averaging all those results (online calculators and spreadsheets make this easy)

The second group, by comparison, has a standard deviation of 1.0488. In other words, the second group-- WITH THE SAME MEAN RADIUS-- has almost twice as much variation as the first group!

Let's assume a "normal" distribution applies to the shot dispersion about the center. We apply the well-established fact that 95% of data points will lie within 2 standard deviations of the mean.

So in the case of the first group, we add two standard deviations (0.5798*2=1.1596) to the mean (1.6) and we get 2.7596 inches as a 95% confidence value.

In other words, based on the first group, we would conclude with 95% confidence that the rifle can hold a group a diameter of ~5.5"-- 95% of shots would lie within a circle of that diameter.


But for the second group (which recall, has the same mean radius), that 95% confidence circle would have a diameter of ~7.4"


Those of you with a military background may be familiar with the concept of "circular error probable" which is the same idea, just applied at 50% confidence instead of 95%. I find 95% to be more useful for my purposes, and it's something of an engineering gold standard.


So there you have it. That's why Mean Radius is not the whole story. It's not just how the *average* distance of the shots from the center of the group, but also scatter in the data points that calculate that average.

You said breif

Quite an analysis, thanks for sharing.

 

[451_Detonics]

Every time I see a post about a rifle's accuracy I feel obliged to post this link...does it apply to most of us casual shooter? Nope...but it is an example of the extremes some folk go to in the name of accuracy...

Secrets of the Houston Warehouse

 

[Hookeye]

I thought it common sense, group size avg involved std dev.
Always has for me, even before I went to eng school.
Consistently good trumps a fantastic group one day vs horrible the next.

BTW, if you lay off shooting for a spell and go back, using 12X or higher mag........some shockingly horrible looking groups might not be as bad as you think LOL.

 

[Hohn]

Every time I see a post about a rifle's accuracy I feel obliged to post this link...does it apply to most of us casual shooter? Nope...but it is an example of the extremes some folk go to in the name of accuracy...

Secrets of the Houston Warehouse

Great read, thanks for posting.