Here is yet another example of just how useful the mean radius can be in evaluating the accuracy of ammunition.  I’ve been doing a comparison of Hornady’s 75 grain TAP FPD load and their NATO pressure 75 grain 5.56 TAP load for a thread in the ammunition forum.  (Much of the information for this post is a dupe, but I felt it was relevant to this thread also.)  As part of that comparison I wanted to see if there was a noticeable difference in the accuracy of the two different loads.  It has been my experience that NATO pressure loads do not shoot nearly as accurately as loads that run 150-200 fps slower, (depending on the weight of the bullet.)

In testing the two different loads I used a 20” Colt HBAR with a NATO chamber, chrome lining and a 1:7” twist.  (I normally use one of my Krieger stainless steel barrels for accuracy testing but I wasn’t going to fire a NATO pressure load through a match chambered barrel.)  In order to ensure that the Colt barrel was capable of detecting a difference in the accuracy of the Hornady loads, I also fired control loads using handloaded Sierra 77 grain MatchKings.  (This control load was not tuned to this particular barrel.  It is one of my “standard” handloads that has proven to shoot well in a variety of barrels with a 1:7” twist.)

All testing was done from a bench at 100 yards and all strings of fire consisted of 10 shots each.  Three 10-shot groups from the control load measured:

1.05”
1.07”
1.10”.

Here is a snapshot of one of the targets of the control load.



Three 10-shot groups of the 75 grain TAP FPD load measured:

1.08”
1.15”
1.18”

This is a snapshot of the smallest group of the TAP FPD load.




Lastly, three 10-shot groups of the 5.56 TAP load measured:

1.16”
1.38”
1.45”.

This is a pic of the best group from the 5.56 TAP load.



For each of the above loads, I compiled a 30-round target by overlaying the three 10-shot groups on each other using the RSI Shooting Lab software program.  This program also gave the mean radius measurement for each of the 30-round composite targets as follows:

Control load = 0.32”

TAP FPD load = 0.37”

5.56 TAP load = 0.46”.

Here are the composite targets for comparison.



This clearly shows:

a.) that the control load was the most accurate load and that the barrel was therefore capable of detecting a difference in the accuracy of the two Hornady loads and

b.) that the 5.56 TAP load is clearly not quite as accurate as the TAP FPD load, although not by much in practical terms.  (We’re talking about a difference of 0.37” between the smallest TAP FPD group and the largest 5.56 TAP group from 100 yards.)

Another interesting point I noticed with the groups from the above test is how well Hatcher’s one-third rule works with those results.  You may recall from my explanation of the mean radius that Hatcher states that the mean radius should measure about one-third of the extreme spread of a group  (or conversely, the extreme spread of the group should be about 3 times the mean radius).

The three 10-shot groups from the Hornady TAP FPD load have an average extreme spread of 1.14”.   The mean radius for that load is 0.37”.  

3 X 0.37” = 1.11”

The three 10-shot groups from the Hornady  5.56 TAP load have an average extreme spread of 1.33” and a mean radius of 0.46”.

3 X  0.46” = 1.38”

Originally Posted By bnorman: Originally Posted By DK-Prof: Standard deviation from what? 3. Even better is mean radius AND standard deviation from that mean, which would help differentiate among groups with the same mean radius, but different distibutions. (Not to speak for Molon, of course, but that is my take on it). I was thinking more along the lines of standard deviation from the center of mass (the vector mean, or the center of the group as defined by Molon).  It would be interesting to fit a large number of shots (100ish) from the same shooter/rifle/range/etc to see how well it fits a 2d gaussian.  Then one could see the comparison of (x0,y0) from the fit with the computed center of the group, how well the tails behave, etc.

Ok bnorman, this post is for you.  I took the 30-round composite targets of the 5.56 TAP and TAP FPD loads described in the above posts and calculated the radial standard deviations of these groups using this formula:



I took me almost twenty minutes to plot the x:y coordinates of all the shots of a single composite target and calculate the radial standard deviation.  (I won’t be doing this very often.)

The TAP FPD load has a radial standard deviation of 0.42”.  The mean radius for the load is 0.37”.

The 5.56 TAP load has a radial standard deviation of 0.50”.  The mean radius for the load is 0.46”.

The mean radius results seem to parallel the radial standard deviation results rather nicely.

(The above radial standard deviations were edited slightly to reflect a better coordinate measuring technique.)

Here is another interesting point I just picked up from a back-issue of American Rifleman.  The radial standard deviation should be roughly equal to 1.11 times the mean radius.

Using the TAP FPD and 5.56 TAP groups from my previous posts as examples the rule bears out rather well.  

TAP FPD

Mean radius of 0.37” X 1.11 = 0.41”, with the actual radial standard deviation being 0.42”


5.56 TAP

Mean radius of 0.46” X 1.11 = 0.51”, with the actual radial standard deviation being 0.50”.

Sooooo......which term describes the maximum distance  from POA I can expect  from any given rifle/load???

Excellent read and very thought provoking thus far.
ta<gg>(ck)ed for later.

--VT

Originally Posted By leebol: Sooooo......which term describes the maximum distance  from POA I can expect  from any given rifle/load???

Maximum shot radius is the term I believe you are looking for.  It is also referred to as maximum group radius or extreme group radius.  

The maximum shot radius “is the distance from a group's statistical center to the center of the most distant hole.”  It is basically your worst shot in the group, the one farthest from the center of the group.  Using the maximum shot radius measurement to scribe a circle around the center of impact of the group will give you a diameter (twice the maximum shot radius) that all the shots of your group will fit in.

Assuming your sights are perfectly zeroed for the distance you are shooting and you are doing everything right in your shooting, your point of aim should be equal to the center of impact of your group.

In the example below, the maximum shot radius = 1.61”.  Using this measurement to scribe a circle around the center of the group gives you a circle with a diameter of 3.22” that all the shots of the group fit in.  Obviously the more shots in a group the larger the maximum shot radius will tend to be.







In the RSI Shooting Lab analysis the maximum shot radius is represented by the outer blue circle.  For example this 10-shot group of the 5.56 TAP load has a maximum shot radius of 0.65”.  Therefore all the shots from the group will fit in a 1.3” circle.  Keep in mind this is not the same thing as the extreme spread, as the extreme spread for this group is only 1.16".  (The X-ring on the actual target below  measures 1.5".)









You could also go the whole six-sigma route with the radial standard deviation, but that is a little much even for me.

Thanks...well done. Have you put this in a doc of pdf that can be saved for reference?

Originally Posted By leebol: Thanks...well done. Have you put this in a doc of pdf that can be saved for reference?

Good suggestion.  I'm actually in the process of doing just that.

Originally Posted By Molon: I just finished reading an excellent article entitled "Is 'Group Size' the Best Measure of Accuracy?" by John E. Leslie.  The article reinforces many of the points made in this thread but goes much further in a comparison of the different statistical methods used to measure a group.   Below is a graph from the article comparing the different methods of measurement including extreme spread, figure of merit, diagonal of dispersion, mean radius and radial standard deviation.  (I believe the radial standard deviation is what  bnorman was referring to in his posts.) Lowest in success rate is the extreme spread and highest is the radial standard  deviation, followed closely by the mean radius. home.comcast.net/~gocartmozart/measurement_comparisons.jpg

The other thing this illustrates is that you get bad data from small sample sizes.  Note how ALL the measurement methods provide equally poor results with small samples.  Of course the problem you run into with larger sample sizes is that it gets harder to identify the individual shots if you're working with an accurate rifle and ammo.  

Originally Posted By spanky02: <center><table width=85% border=0><tr><td width=100% class=textQuote><hr height=1px color=black noshade>Originally Posted By Molon: I just finished reading an excellent article entitled "Is 'Group Size' the Best Measure of Accuracy?" by John E. Leslie.  The article reinforces many of the points made in this thread but goes much further in a comparison of the different statistical methods used to measure a group.   Below is a graph from the article comparing the different methods of measurement including extreme spread, figure of merit, diagonal of dispersion, mean radius and radial standard deviation.  (I believe the radial standard deviation is what  bnorman was referring to in his posts.) Lowest in success rate is the extreme spread and highest is the radial standard  deviation, followed closely by the mean radius. home.comcast.net/~gocartmozart/measurement_comparisons.jpg <hr height=1px color=black noshade> The other thing this illustrates is that you get bad data from small sample sizes.  Note how ALL the measurement methods provide equally poor results with small samples.  Of course the problem you run into with larger sample sizes is that it gets harder to identify the individual shots if you're working with an accurate rifle and ammo.

Good point.  That's one of the reasons that I started using 3 seperate 10-shot groups and overlaying them on each other using the RSI Shotting Lab software program for a 30-round composite target.

To Molon and et al who participated in gathering the data,

Thank you very much for your effort and the information. This is definitely one of those threads that needed to be tacked or moved permanently to the FAQ section.




...so does this mean it is okay with my 3.5" 10 shot groups at 100 yards?


I shoot my AR15's with the magazine as a support, use Q3131 and in prone. Iron sights or TA31. I was wondering why my groups were so bad compared to the other people here... The absolute best I could do was about 1.5" and that was on a VERY lucky day.

This is why, when reporting group size, I usually name not only how many rounds were fired, but whether or not the group was typical for the day.

For example, with Q3131A and my 20" with irons, one day I shot 3 rounds each at three un-marked silhouettes at 100 yards; 3 on one, 3 on the next, and so on.  I fired supported by one of my ammo cans.  One was a little under 2", and both of the others were something under 2.5"

Was each group on the same location on each target?  No.  The silhouettes were carved by hand out of white cardboard, and were not even the same size.  The irons are zeroed for 300M, so I was holding a bit low, and I had no actual "spot" on the target to aim at consistently.  And to top it off, most of my other shooting that day, in groups of 3, 6, etc., was around 3"

But I was still pretty happy with myself.  

The rationale for overlaying three 10-shot groups is in line with statistical methods for sample sizes.  In statistics using a sample size of 30 is considered a large sample size and gives you a more accurate distribution.

Outstanding thread/information. Thanks to all who contributed, it was a thoroughly enjoyable read!

Ok, I am not a math guru here so I won't even go into that side.  You are probably right about the average.  

Having said that, when you get past 3 rounds, you get into other factors like Heat.  Heat on a barrel can have a HUGE effect...maybe not on a heavy barrel but I know for a fact that several of my bolt rifles will shoot sub inch groups from a cold barrel...but if that barrel is very hot, as in after 5 shots, 95 deg outside, no cooling between shots- the groups will open up to about 1.5 inches or more.  

I think the cold bore group is more realistic since that would likely be the shots fired at game.  

Today, I was working with my 300 Mag and it will consistently shoot 3 rounds of factory remington Core Lock into .75 of an inch center to center from a cold bore.  

I tried 2 five shot groups in a row, then shot another 3 round group.  The barrel was too hot to hold.  The group was 1.75 inches center to center.    Heat was the only thing that changed.

When you get beyond 3 rounds, the heat becomes a factor and I don't think heat should be part of it.  If heat is the deciding factor, a heavy barrel will always win and weight is not as important for what I do as portability.  

Good post.

Personally, I reserve all three shot groups for intruders...

Originally Posted By FedDC: Ok, I am not a math guru here so I won't even go into that side.  You are probably right about the average.   Having said that, when you get past 3 rounds, you get into other factors like Heat.  Heat on a barrel can have a HUGE effect...maybe not on a heavy barrel but I know for a fact that several of my bolt rifles will shoot sub inch groups from a cold barrel...but if that barrel is very hot, as in after 5 shots, 95 deg outside, no cooling between shots- the groups will open up to about 1.5 inches or more.   I think the cold bore group is more realistic since that would likely be the shots fired at game.   Today, I was working with my 300 Mag and it will consistently shoot 3 rounds of factory remington Core Lock into .75 of an inch center to center from a cold bore.   I tried 2 five shot groups in a row, then shot another 3 round group.  The barrel was too hot to hold.  The group was 1.75 inches center to center.    Heat was the only thing that changed. When you get beyond 3 rounds, the heat becomes a factor and I don't think heat should be part of it.  If heat is the deciding factor, a heavy barrel will always win and weight is not as important for what I do as portability.

Not to be obtuse, but if heat is an issue in your barrel, why would that prevent you from firing 10 shots, in VERY slow succession, allowing the barrel to cool between them - IF you were interested in really assessing the barrel's accuracy?

After re-reading some of the posts in this thread I decided to revisit the issue of the accuracy of 55 grain FMJ bullets.  While disputing the wild internet claims of minute of angle accuracy using Federal’s XM193 was the original impetus for this thread I now wanted to find out just what level of accuracy the 55 grain FMJ bullet was capable of delivering.  

All shooting for this test was done at 100 yards from a bench rest.  All groups consisted of ten shots each.  For the first part of this test I decided to use the factory load of IMI M193.  It is the most consistently accurate M193 load that I have come across.  For the second part of the test I wanted to see how much I could improve on the accuracy of the IMI M193 load using hand-loaded 55 grain FMJ bullets.  I choose to hand-load Hornady’s 55 grain FMJ-BT bullet for the same reason I used the IMI M193; it is the most accurate 55 grain FMJ bullet I have come across.  The bases of these bullets appear to be much more uniform than the comparable bullet from other manufacturers.  While I have obtained some singularly good groups using Hornday’s 55 grain bullet in the past, for some reason I have never created a 30-round composite group using this bullet.  

Three 10-shot groups of the IMI M193 were fired from a free-floated 20” Colt HBAR with a 1:7” twist that is chrome lined and has a NATO chamber.  (The Colt barrel was used because I’m just not willing to risk pressure issues by firing mil-spec M193 through one of my match chambered Krieger barrels.)  Prior to obtaining those groups I fired a control load consisting of hand-loaded 77 grain Sierra MatchKings that measured 1.10”.  The three groups of IMI M193 measured:

2.24”
2.59”
2.62”.

The three groups were overlayed on each other using RSI Shooting Lab software to form a 30-round composite group and the mean radius measured 0.72”.

Next, I obtained three 10-shot groups of the hand-loaded Hornady 55 grain FMJ bullets fired from a free-floated 20” Krieger HBAR with a 1:9” twist.  This hand-load runs about 200 fps slower than the mil-spec M193 loads.  I also fired a control group from this barrel using hand-loaded 55 grain Sierra BlitzKings that measured 0.94”.  The three groups of Hornady 55 grain FMJ loads measured:

1.66”
1.72”
1.74”.

As before a composite 30-round group was created and the mean radius measured 0.54”; not match quality ammo, but a definite improvement over M193 and I’ll dare to say probably about as good as 55 grain FMJ bullets are going to shoot consistently.

Here are the 30-round composite groups side-by-side for comparison.  I’ve also included a 30-round composite group of hand-loaded Sierra 52 grain MatchKings fired from the Krieger barrel for additional comparison.  (The three 10-shot groups measured 0.88", 0.89" and 0.88".)





P.S.

At one point during the testing of the Hornady 55 grain FMJ load, my inner-Internet-Commando asserted itself and forced me to fire a 3-shot group of the 55 grain Sierra BlitzKing control load.   It measured 0.214”.

Excellent stuff (as usual).  Thanks Molon!


It seems like there is just no way for someone to consistently shoot MOA or less with 55 gr. FMJ - and certainly not anywhere ever remotely close to MOA with XM193.

You guys are shooting great, i would be happy if i could get there!

Here is some more data on the accuracy of 55 grain FMJ bullets.  I recently fired an exceptional group for a 55 grain FMJ bullet.  The group consisted of 10-shots fired from 100 yards using hand-loaded Hornady 55 grain FMJ bullets.  The group measured 1.44” and was fired from a free-floated 20” government profile barrel with a NATO chamber and chrome lining.  This singular group was actually better than the three groups I fired from the Krieger stainless steel match barrel used for the testing reported in my post 3 posts above.

As previously stated this is an exceptional group for a 55 grain FMJ bullet.  The interesting point about this is that the 10-shot group before it measured 2.18” and the 10-shot group after it measured 2.43”.  Also, I was not able to reproduce the 1.44” group during the rest of the shooting with that ammo that day.

I overlayed the three groups from above on each other using RSI Shooting Lab software and obtained a 30-round composite group with a mean radius of 0.60”.   Even including the 1.44” group, the composite mean radius of the three groups was still larger than the mean radius from the three 10-shot groups of Hornady 55 grain FMJ hand-loads mentioned in my above post that measured 1.66”, 1.72” and 1.74”.  The composite mean radius for those groups was 0.54”.

This shows that even though we can occasionally obtain a singularly good group from a particular load and rifle, it is not truly indicative of the accuracy of the rifle/ammunition combination.  That group is the exception rather than the norm.  This also shows again just how useful the mean radius can be in evaluating the accuracy of a given rifle/ammunition combination.

As I stated earlier in this thread it’s not that 55 grain bullets per se are incapable of MOA accuracy, it’s the full metal jacket variety that are not capable of that level of accuracy.  To see what a match quality 55 grain bullet is capable of from an AR-15, I loaded up some Berger 55 grain Match hollow points and headed to the range with my Krieger barreled AR-15.

Using my usual method of accuracy testing from a bench-rest at 100 yards, I obtained three 10-shot groups of the Berger 55 grain hand-load for analysis.  The three groups measured 0.71”, 0.97” and 0.97”.  More importantly, when those three groups were overlayed on each other using RSI Shooting Lab software the Berger load had a mean radius of 0.25” for the 30 round composite target.  A composite mean radius of 0.25” for a 30-round composite group is right at the limits of my shooting ability with an AR-15.  (In other words these bullets are most likely capable of even greater accuracy.)

Here is a pic of the best 10 shot group from the Berger load.





Here is the 30-round composite group form the Berger load compared to a 30-rond composite group of hand-loaded Hornady 55 grain FMJ bullets.

When I go to the range now, I cannot resist firing off a 10-shot group, and then a couple of 3-shot groups, just to provide more illustrations of the basic point Molon was making in the original post in this thread.



Okay, so sure, I technically fired 12 shots in my 10-shot group (counting to 10 is hard ) - but neither of the two last shots contributed to the extreme spread (if anything, they helped mean radius).

Once again, it is clear to see that if I had JUST fired the 3-shot groups (especially the top one), I would incorrectly believe that I was a 1 MOA shooter with this rifle/ammo.  Also, once again, if one OVERLAID the two separate 3-shot groups, it also illustrates nicely that the accuracy of the rifle/ammo is not as good as each single group seems to indicate.

...

Originally Posted By Molon: Another interesting point I noticed with the groups from the above test is how well Hatcher’s one-third rule works with those results.  You may recall from my explanation of the mean radius that Hatcher states that the mean radius should measure about one-third of the extreme spread of a group  (or conversely, the extreme spread of the group should be about 3 times the mean radius). The three 10-shot groups from the Hornady TAP FPD load have an average extreme spread of 1.14”.   The mean radius for that load is 0.37”.   3 X 0.37” = 1.11” The three 10-shot groups from the Hornady  5.56 TAP load have an average extreme spread of 1.33” and a mean radius of 0.46”. 3 X  0.46” = 1.38”

***************************************************



So, the above group, the one on the right, measures 2.5" across, its mean radius would be .83", and the group 'size' would be twice that, or  1.6".

I  feel better about it already!





... and the far right, middle, measuring 1.25" across, has a mean radius of .42", and a mean spread of .84" ... at 100 yards, with a nasty mirage (heat shimmy), that 's not bad.

Originally Posted By FrankSymptoms: Originally Posted By Molon: Another interesting point I noticed with the groups from the above test is how well Hatcher’s one-third rule works with those results.  You may recall from my explanation of the mean radius that Hatcher states that the mean radius should measure about one-third of the extreme spread of a group  (or conversely, the extreme spread of the group should be about 3 times the mean radius). The three 10-shot groups from the Hornady TAP FPD load have an average extreme spread of 1.14”.   The mean radius for that load is 0.37”.   3 X 0.37” = 1.11” The three 10-shot groups from the Hornady  5.56 TAP load have an average extreme spread of 1.33” and a mean radius of 0.46”. 3 X  0.46” = 1.38” *************************************************** i56.photobucket.com/albums/g169/franksymptoms/Gun%20stuff/Targets8-12b.jpg So, the above group, the one on the right, measures 2.5" across, its mean radius would be .83", and the group 'size' would be twice that, or  1.6". I  feel better about it already! i56.photobucket.com/albums/g169/franksymptoms/Gun%20stuff/Targets8-12a.jpg ... and the far right, middle, measuring 1.25" across, has a mean radius of .42", and a mean spread of .84" ... at 100 yards, with a nasty mirage (heat shimmy), that 's not bad.

FrankSymptoms,

Hatcher's one-third rule only applies to 10-shot groups.  Do the squares on your target measure one inch?

Molon

Originally Posted By Molon: FrankSymptoms, Hatcher's one-third rule only applies to 10-shot groups.  Do the squares on your target measure one inch? Molon

Yes, they are 1" squares. I generated them using MS Excel.

I'll go out and shoot some 10-shot groups tomorrow. It's been raining so maybe I won't be dealing with heat shimmy too!

Originally Posted By FrankSymptoms: Originally Posted By Molon: FrankSymptoms, Hatcher's one-third rule only applies to 10-shot groups.  Do the squares on your target measure one inch? Molon Yes, they are 1" squares. I generated them using MS Excel. I'll go out and shoot some 10-shot groups tomorrow. It's been raining so maybe I won't be dealing with heat shimmy too!

Post a couple of your 10-shot groups and I'll see what I can do to analyze them.  I'm not quite sure how you came up with those measurements from your first post.

Molon

PS  Do you work in health care? "FrankSymptoms"  

Fantastic post. I am brand new here and have have a quick question. I want to send in my dues to become a member but they ask that you include you "board name" with payment. Sorry to interupt this thread with a dumb question but is you "board name" the same as your user name? Thankxx

Originally Posted By devious-1: Fantastic post. I am brand new here and have have a quick question. I want to send in my dues to become a member but they ask that you include you "board name" with payment. Sorry to interupt this thread with a dumb question but is you "board name" the same as your user name? Thankxx

Yes.  Welcome to the board.

Originally Posted By Molon: PS  Do you work in health care? "FrankSymptoms"

Not in health care. I just picked it because it sounded funny to me and I'd just read the term in a book. It was (one of) my AOL screen name(s) lo, these many years ago.

Not very many people comment on it.

Molon, here are some 10-shot groups I made. These were on a hot day, with lots of mirage shifting.

Here's an interesting site that has a section on the 'flyers' that destroy your otherwise perfect sub-minute-of-Sunday groups. He has a lot of other interesting stuff on his site, too.

Tech Tips

Basically, he's saying that the reason you get flyers is that the bullets themselves, the slugs, are not always perfectly formed! So if one bullet is rounder (and shorter!) than the others, it will contact the rifling sooner!

{edited: Probably will fly differently too, due to the different aerodymanics.}

I found this out just last night! Some of my bullets were a few thousandths of an inch longer than the others... AFTER they were loaded. So: if one slug was "fatter" that another, it would have contacted the seating die sooner.

Originally Posted By FrankSymptoms: Molon, here are some 10-shot groups I made. These were on a hot day, with lots of mirage shifting. i56.photobucket.com/albums/g169/franksymptoms/Gun%20stuff/best-score.jpg

Now we've got something to work with!  Using your better group on the right hand side of the target I calculated the mean radius and extreme spread of the group using RSI Shooting Lab.  The mean radius measures 0.58".






Using Hatcher's Rule we can calculate that the extreme spread (group size) should measure approximately 1.74".

3 X 0.58 = 1.74"

The actual extreme spread of your group is 1.72".

Well, I GUESS that's OK. <2" is OK I suppose. I'd hoped I would get better results than that though.

I have a few 20-round baggies of .223 that I've handloaded so maybe I can improve on that a little...

Originally Posted By FrankSymptoms: Here's an interesting site that has a section on the 'flyers' that destroy your otherwise perfect sub-minute-of-Sunday groups. He has a lot of other interesting stuff on his site, too. Tech Tips Basically, he's saying that the reason you get flyers is that the bullets themselves, the slugs, are not always perfectly formed! So if one bullet is rounder (and shorter!) than the others, it will contact the rifling sooner! I found this out just last night! Some of my bullets were a few thousandths of an inch longer than the others... AFTER they were loaded. So: if one slug was "fatter" that another, it would have contacted the seating die sooner.

Quality open-tip-match bullets such as Sierra’s MatchKing bullets can easily vary in length by over 10 thousandths of an inch.  This is a byproduct of the manufacturing process used for open tip bullets and it has no measurable effect on accuracy at say 100 yards when fired from an AR-15.  

I performed a test in which I hand-loaded thirty 52 grain Sierra MatchKings that had been sorted so they all had the same length and then fired them in three 10-shot groups from 100 yards from an AR-15 with a Krieger SS match barrel.  I then compared those results to the results of firing three 10-shot groups of thirty hand-loaded 52 grain Sierra MatchKings randomly chosen from the same lot.  There was no discernable improvement in accuracy with the bullets that had been sorted to the same length.

The measurement you need to look at to see how consistently OTM bullets have been manufactured is the base-to-ogive length.  It is the beginning of the ogive section of the bullet that first comes into contact with the lands of a barrel.  The variance for this measurement with quality OTM bullets will be far smaller than the variance of the overall length of the bullet.  There are several companies offering devices that are used to obtain this measurement.  

With a loaded round, you can measure the length from the base of the case to the ogive of the bullet using an RCBS Precision Mic.  Using the Precision Mic, I checked the base-of-the-case-to-bullet-ogive measurement of 10 rounds of hand-loaded 77 grain Sierra MatchKings.  The measurement only varied by plus-or-minus 0.001”.

A Sierra 77 grain MatchKing loaded to magazine length is going to have a bullet jump of over 150 thousandths of an inch to reach the lands in a NATO chambered Colt barrel.  Even in one of my match chambered Krieger barrels the 77 grain MatchKing still has a jump of 0.070”.  That plus-or-minus 0.001” difference in bullet jump due to the variation of the bullet ogive is not going to have any demonstrable effect on accuracy when fired from an AR-15.

Originally Posted By GHPorter: I want to know more about that machine rest.  How about more pictures, a web site, anything!

I came across another picture of Rick Jamison's machine rest.

Here is an interesting quote by John Feamster from a chapter in Precision Shooting, Reloading Guide.

"The US Army Marksmanship Unit at Ft. Benning, GA uses a minimum of 3 consecutive 10-shot groups fired with the rifle in a machine rest when testing service rifles."

Hilarious post - I’m forwarding this link to all my gun-friends that aren’t ARFCOM’ers!

Great post thanks

After reading this whole thread I've come to 3 conclusions:

1.  Molon really knows what he's doing when it comes to testing ammo.  (DK-Prof too.)

2.  I used to think that when my 5-shot groups varied a lot in size and/or POI that it was poor technique on my part.  Thanks to you guys I can now blame statistics instead.  

3.  ES is almost as good as the fancy stuff if you use 10-shot (or more) groups

I'll get back to #3, but first a question:
My SOP for ammo testing in the past has been to shoot 4-6, 5-shot groups and then average the sizes of all the groups, comparing the average for different loads.  Is that statistically valid?  Does it really tell me anything useful?

Back to conclusion #3:
You've noted several times in this thread that Hatcher's 3x rule holds very closely for the groups you've analyzed, with ES for the groups being very close to 3x the mean radius.  Reversing that, if I measure ES of a suitably large (in # of shots) group and divide by 3 then I should get a number that's very close to the actual MR, without doing all the work of actually computing it.  Close enough for comparing loads for meaningful differences, right?
Actually, assuming I'm just comparing two loads I should be able to just compare the ES of the two groups, since the MR and ES differ by a constant.
Provided that I shoot all 20 or 30 shots to the same target or overlay the groups to measure ES for the 20/30-shot composite.


If I'm understanding mean radius right, I can expect shots to be within that distance of POA half the time and outside half the time?  And ES/2 to be a approximately a worst-case radius?   Is there an easy way to get to radial std. dev. that would give a radius that I can count on for, say 95% of the shots to be inside of?

Originally Posted By Mike_L: 1.  Molon really knows what he's doing when it comes to testing ammo.  (DK-Prof too.)

To be fair, Molon really DOES know what he's doing, whereas I'm more like a trained monkey that goes to the range and shoots a little, and then asks Molon to explain it to me.

If you already have had a new trigger group of match quality or better or trigger job, Great. If not get it.  Second, let your barrel cool down 10 to 15 min. between groups.  Now, if you're doing that start cleaning your barrel in between cool downs.  This will assist you in maintaining your shot groups.

Hi Mike_L,

Thanks for reading and replying to this thread.  Since you have several questions I'd  like to respond to them individually, but I'm a little tied up with testing for my Hornady TAP ammunition thread at the moment.  I'll get back to you with some answers soon.

Molon

I'd just like to put in another word for Molon; I'm taking another statistics course right now (my last one was about 12 years ago and frankly I'm rusty anyway) and his data-based approach to significance is very helpful.  The minimum group size is directly related statistically to the quality of the statistics you calculate from it.  More shots means more data-and higher quality data too.  Thanks for some reality in what is often a very, VERY dry subject.

Originally Posted By Mike_L: 3.  ES is almost as good as the fancy stuff if you use 10-shot (or more) groups

Not really.  The beauty of using the mean radius to analyze the accuracy of a group (technically the precision) is that each and every shot in the group is used as a data point.  When using the extreme spread only two shots from the entire group (no matter how many shots in the group) are used for analysis and they are the two worst shots of the group to boot.  

Analyzing a 10-shot group using the mean radius gives you 10 data points to work with.  Analyzing the same 10-shot group using the extreme spread only gives you 2 data points to work with.  10 data points is better than 2 data points hands down.  Now if you are going to use just the extreme spread to analyze a group, 10-shot groups are far more useful than 3-shot or 5-shot groups.

I'll get back to #3, but first a question: My SOP for ammo testing in the past has been to shoot 4-6, 5-shot groups and then average the sizes of all the groups, comparing the average for different loads. Is that statistically valid? Does it really tell me anything useful?

Probably not.  If you were to shoot a large sample of 5-shot groups in a row (say 30 groups) and count every group and every shot in every group (none of this calling a shot a “flyer” because Jupiter wasn’t aligned with Mars when you fired the shot) then you would have some useful data for comparison.  Multiplying the average extreme spread of a large sample of 5-shot groups by 1.25 will give you an expected average extreme spread of a large sample of 10-shot groups.

Back to conclusion #3: You've noted several times in this thread that Hatcher's 3x rule holds very closely for the groups you've analyzed, with ES for the groups being very close to 3x the mean radius. Reversing that, if I measure ES of a suitably large (in # of shots) group and divide by 3 then I should get a number that's very close to the actual MR, without doing all the work of actually computing it. Close enough for comparing loads for meaningful differences, right? Actually, assuming I'm just comparing two loads I should be able to just compare the ES of the two groups, since the MR and ES differ by a constant. Provided that I shoot all 20 or 30 shots to the same target or overlay the groups to measure ES for the 20/30-shot composite.

Not exactly.  Hatcher’s rule will give you a reasonable estimate of the expected average extreme spread of 10-shot groups if you know the mean radius and then usually only if the shot groups have a normal distribution.  It doesn’t necessarily hold true in the opposite direction or even always in the intended direction.

Allow me to illustrate.  Suppose I send you a box of ammunition for you to test in your favorite AR-15.  Like-wise, I test this same lot of ammunition in my favorite AR-15.  You post the results of your 10-shot group on line and report the extreme spread to be 2.1”.

I also post the results of my 10-shot group on line and quite amazingly my group also measures 2.1”.

Now, without actually seeing either of the two targets, we would assume that they are essentially equal in accuracy.  We might even try to use the reverse of Hatcher’s rule and calculate the mean radius of each of our groups to be 0.70”.  Unfortunately we would be mistaken.

Here is a picture of your fictional target.  The mean radius measures 0.43”, (not 0.70” as we expected it to be.)





Now here is a picture of my target.  The actual mean radius measures 0.78” ( a lot closer to the expected mean radius of 0.70”, but still not quite what we expected it to be.)




Once again the difference in results is due to the difference in the number of data points used in the different measuring methods: the mean radius using 10 data points and the extreme spread only two data points.  With the extreme spread method we just aren’t accounting for all the data in between the two points forming the extreme spread.

If I'm understanding mean radius right, I can expect shots to be within that distance of POA half the time and outside half the time?

With groups having a relatively normal distribution, on average yes.

And ES/2 to be a approximately a worst-case radius?

Negative.  The “worst case radius” is defined by the concept of Maximum Shot Radius which is described in detail on page 7 of this thread.

Is there an easy way to get to radial std. dev. that would give a radius that I can count on for, say 95% of the shots to be inside of?

I have yet to find a software program that will calculate a true two-dimensional radial standard deviation from the statistical center of a shot group following this formula:




Some target analysis programs give a standard deviation number, but it is only the standard deviation from the mean radius (which I haven’t found a lot of use for), not a true two dimensional radial standard deviation.  Multiplying the average mean radius of 10-shot groups by 1.11 will give you an estimate of the average radial standard deviation and then of course multiplying that number by two will give you your 95% confidence level radius.


Molon