The real statistics of accuracy and tuning

[tachyon]

So,

Over here viewtopic.php?f=5&t=15627 @DingoDeerHunter suggested it is not possible to tune a rifle based on this video: https://www.youtube.com/watch?v=JSxr9AHER_s called "Science agrees: 5-shot groups are pointless"

I commented that the observation from Blackburn Defense that bullets do not hit the target according to a normal distribution is correct, but the rest of the Blackburn analysis is increasingly wrong. I noted, quite reasonably, that anyone who is shooting 4-8" groups (1-2 hands wide) @ 100 yards with both 22LR and Centerfire almost certainly needs to work on their marksmanship.

I then went on to try to explain (in layman's terms) why it is relatively easy to pick "out of tune" because the groups are big, and also observe the "in tune" area where groups are much smaller. Indeed I shared a target picture...

I’m sorry you don’t understand the significance of the fact that the model is not a normal distribution and that you believe anecdotal evidence is significant.

This is like a kind of religion, and no amount of evidence will persuade y’all.

Be safe and enjoy

At the risk of ignoring “Never argue with an idiot. They will drag you down to their level and beat you with experience.” ― Mark Twain

If anyone's interested I'm happy to work through the statistics of dispersion that show the following:

# Yes, outliers exist...

dispersion1.png

# Yes, bullets do NOT hit the target in a normal distribution

dispersion2.png

# Yes, the more bullets you shoot the bigger the group > BUT < the more "accurate" your estimate of accuracy is

dispersion3.png

# Yes, more bullets do give a more accurate result, but there are diminishing returns

dispersion4.png

# Here we get to the crux of the matter - a gun that shoots well, shoots well most of the time and produces a (generally) small group

dispersion5.png

Unfortunately there seems to be a 5 image limit for uploads so the most interesting image is missing - namely the overlap between 2,3,5,10,20,40 shot groups for real (random) data that matches 2 rifles - one that shoots 1SD and the other that shoots 1/2SD.

If anyone wants the R code that generated this you are welcome.

[tachyon]

# And here is how many shots you need in your group to differentiate between SD and 1/2SD

dispersion7.png

[DingoDeerHunter]

Blackburn Defense showed you can distinguish two rifles, because small samples will show a relevant p value for large differences. So your example is not apposite. What we are talking about is what sample size is required to obtain a p value for small differences between powder, bullet seating and Brock’s polarizer magical devices. They have demonstrated the p values as meaningless.

As for 4” to 8” group comment - what distance do you think he was shooting, it wasn’t 100 yards.

The further articles I posted in the original thread showed that 10 x 5 shot groups using the same load etc had a difference between smallest group and aggregate of a factor of nearly 3. So if you’re using 1 5 shot group to test a load or seating depth, it could show up (for eg) as a 1/2moa group or a 1 and 1/12 MOA group depending on nothing other than randomness. They shot using a mechanical system thus no shooter error.

So you do a test of 10 loads and you choose a spot where 3 groups appeared smaller and yet this is more likely to be nothing more than statistical variance.

As for your snarky ad hominem - science doesn’t care about your feelings.

Good shooting

[jcinsa]

I would like someone to explain to me how in the first post on this thread, the first 2 graphs are made using the same data.
It doesn’t seem logical to me, or am I missing something ?
Thanks in advance.
John

[AlanF]

..I commented that the observation from Blackburn Defense that bullets do not hit the target according to a normal distribution is correct, but the rest of the Blackburn analysis is increasingly wrong...

Well that's interesting. I disagree with both Mr tachyon and Mr blackburn . I think that bullets DO hit the target approximately according to a normal distribution. Target faces are two dimensional so to fully describe their positions you need coordinates, not just a radial distance from the centre of the group. If you take the horizontal coordinates of shots, do a frequency distribution table, and plot it, it will resemble a normal distribution bell curve centred on the middle of the group. Same thing for vertical coordinates. This makes sense, because dispersion of shots is caused primarily by the combined effect of a multitude of random errors which act either horizontally or vertically. There will be some systematic errors from bad technique etc. and occasional gross errors from blunders, but in my opinion for the people in our sport who are dedicated enough to do accuracy testing, they are usually insignificant.

[tachyon]

I would like someone to explain to me how in the first post on this thread, the first 2 graphs are made using the same data.
It doesn’t seem logical to me, or am I missing something ?
Thanks in advance.
John

Hi John,

I think what you are asking is "If it is actually less probable that a bullet hits dead centre why does the middle look denser than the edges?"

Hopefully the reason is explained by this image

Screen Shot 2023-06-19 at 12.42.12 pm.png

Dead centre we need 1 bullet to create "full density". At 1 bullet bullet diameter out from here we need 6 bullets to create full density, at 2 diameters out we need 12 bullets to create full density, etc.

Another way of looking at it is this:

If we have 2 factors (a and b) impacting accuracy, both normally distributed around the bullseye then the only way to score a dead centre is for one of the following conditions to apply:

a = 0, b = 0
a = -x, b = +x
a = +x, b = -x

As in each case a+b = 0.

So, while a = 0 is the most common case, b != 0 is far more common than b = 0 for the partner distribution and thus the summation of these 2 "eats" the middle. As we move left/right from the middle it becomes ever more unlikely that a -x will have a simultaneous matching +x to deliver a bullseye.

The most useful things to know from all this is that group size does follow a gently skewed normal distribution and over ~5 shots there is very little gain - a rifle that's shooting 1/4 MOA will have declared itself, just as one shooting 1 MOA will.

one-vs-quarter-moa.png

[tachyon]

..I commented that the observation from Blackburn Defense that bullets do not hit the target according to a normal distribution is correct, but the rest of the Blackburn analysis is increasingly wrong...

Well that's interesting. I disagree with both Mr tachyon and Mr blackburn . I think that bullets DO hit the target approximately according to a normal distribution. Target faces are two dimensional so to fully describe their positions you need coordinates, not just a radial distance from the centre of the group. If you take the horizontal coordinates of shots, do a frequency distribution table, and plot it, it will resemble a normal distribution bell curve centred on the middle of the group. Same thing for vertical coordinates. This makes sense, because dispersion of shots is caused primarily by the combined effect of a multitude of random errors which act either horizontally or vertically. There will be some systematic errors from bad technique etc. and occasional gross errors from blunders, but in my opinion for the people in our sport who are dedicated enough to do accuracy testing, they are usually insignificant.

Hi Alan,

While I agree the result is a bit counter intuitive, it is correct.

Please see the attached file which is the full R code that generated the results inline with the results.

As you will see, they were generated from 2D co-ordinate positions and a model with 3 degrees of freedom. As you will also see from the histograms of X and Y these x,y co-ordinates are quite clearly normally distributed.

Screen Shot 2023-06-19 at 1.55.18 pm.png

What is NOT normally distributed is their impact positions on the target when these 2D co-ordinates are reduced to radial displacement from the centre aimpoint.

Screen Shot 2023-06-19 at 2.01.22 pm.png

This: viewtopic.php?f=5&t=15634&p=112042#p112041 explanation for jcinsa may help you visualise what is going on.

Bullet Dispersion.html.zip

[AlanF]

I would like someone to explain to me how in the first post on this thread, the first 2 graphs are made using the same data.
It doesn’t seem logical to me, or am I missing something ?
Thanks in advance.
John

John,

Again I have to disagree with Mr tachyon. The second graph is not relevant to shot density as it based only on radial distance to the shots. If the first graph was thinned out using less data, it would show that. Typically there is no dip in shot density near the centre of a group. It does however even out near the middle, corresponding with the top of a normal disribution bell curve.

[AlanF]

..I commented that the observation from Blackburn Defense that bullets do not hit the target according to a normal distribution is correct, but the rest of the Blackburn analysis is increasingly wrong...

Well that's interesting. I disagree with both Mr tachyon and Mr blackburn . I think that bullets DO hit the target approximately according to a normal distribution. Target faces are two dimensional so to fully describe their positions you need coordinates, not just a radial distance from the centre of the group. If you take the horizontal coordinates of shots, do a frequency distribution table, and plot it, it will resemble a normal distribution bell curve centred on the middle of the group. Same thing for vertical coordinates. This makes sense, because dispersion of shots is caused primarily by the combined effect of a multitude of random errors which act either horizontally or vertically. There will be some systematic errors from bad technique etc. and occasional gross errors from blunders, but in my opinion for the people in our sport who are dedicated enough to do accuracy testing, they are usually insignificant.

Hi Alan,

Please see the attached file which is the full R code that generated the results inline with the results. As you will see, they were generated from 2D co-ordinate positions and a model with 3 degrees of freedom. As you will also see from the histograms of X and Y these x,y co-ordinates are quite clearly normally distributed. What is NOT normally distributed is their impact positions on the target.

Bullet Dispersion.html.zip

The key thing is what you mean by "impact positions". I'm asserting that the second graph is based on radial distances only to the shots, so does not describe the positions fully. And I don't think its useful, because it implies that shot density on the two dimensional target face lessens near the middle, which typically it doesn't.

[tachyon]

Hi AlanF,

Here is a more detailed rendering of what is happening...

Screen Shot 2023-06-20 at 12.50.46 pm.png

Screen Shot 2023-06-20 at 12.51.22 pm.png

Thanks for Muneeb for the interactive dispersion model.

BulletDispExplanation.html.zip