Originally Posted By Molon: These old back issues of American Rifleman (1950s, 1960s) make for some engaging reading.  Here is a pic from a July 1958 article pointing out why even 5-shot groups are misleading for evaluating the accuracy of ammunition.

Good info - thanks!  I've used 5 shot groups often, looks like I'm going to have to upsize them.

Tag

Originally Posted By PanzerMK7: This is one of the best threads i've ever read here, great job. Pay no attention to the people who can't grasp the concept, darwin will take care of them.

+1

Mike

Thanks again for the compliments.

Great post, thanks for sharing.

Awesome thread, thanks for all the work Molon. 10 shot groups for me I guess...(well unless I need to impress someone I suppose)

I shoot 28 shot groups.  The rationale is to measure the A.D.D. factor.

Wondering what you guys think about  focus fatigue.  I'm not talking about B.R.A.S.S. ...I'm talking about mental focus and awareness.

I have found my 28 shot groups actually tighten up when I shoot them with a 4-5 second cadence while breathing rather than taking my time and trying too hard.  

Originally Posted By Molon: I was curious to hear if you were using XM193.  The mean radius of your group at 100 meters is 0.91" and adjusted for 100 yards it comes out to 0.83".  In a previous post on this page I showed a 30-round composite target I obtained using XM193 fired from 100 yards.  The mean radius of that composite group was 0.88".  It's kind of scary how we both ended up with basically the same results for XM193 when measured using the mean radius.

In case you were interested in another (limited) piece of data on XM193, here's a target I shot today at 100 yards.  Unfortunately, it's not 30 or more rounds, but it is 17 (I was using up the rest of a magazine).  

In part I am just posting it because I was quite pleasantly surprised at the performance of XM193 - the spread looks like its around 2 1/4" and it was a pretty windy day (probably about 20 - 25mph, and gusty).



Of course, I'm also posting it in case you have the patience to run it through your "mean radius" calculator.  

Good eye!  The extreme spread of your group measures 2.27”.  The mean radius is 0.67”.   If you can, the next time you go shooting, fire three 10-shot groups of your XM193 from 100 yards and post your results.  I can then make more of an “apples to apples” comparison.





What lot number is the XM193 you used for this group?  Some lots definitely seem to be more accurate than others.  The limiting factors to the accuracy of XM193 are obviously a.) the bullet, but also I believe b.) the velocity to which the round is loaded.  In my personal experience I have found that 55 grain FMJ bullets fired from an AR-15 tend to be the most accurate at velocities that are downloaded about 200 fps from M193 velocities.  (Not that 55 grain FMJ bullets are known for stellar accuracy to begin with.)*

I have noticed that while older lots of XM193 have the asphalt bullet/case neck sealant, many of the newer lots do not.  I wonder if the sealant has a negative effect on accuracy?

Below is a pic of the best 10-shot group from 100 yards that I have obtained using handloaded Hornady 55 grain FMJ bullets.  The extreme spread is 1.38” and the mean radius is 0.41”.  I don’t know if 55 grain FMJ bullets are capable of anything more accurate than this.




*I love this quote from the book Black Magic:

For accuracy usage, the 55 grain FMJ is an evil bullet... Under sufficient magnification, it wouldn't surprise me a bit to find a small "666" tattooed into its shiny little ogival head. Abandon all hope, ye who seek tiny groups here!

Some great info. Thanks to all we've contributed.

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

I've finished going through all my back issues of Shooting Times and I haven't come across any more pictures of Rick Jamison's machine rest.  However, I did just recently come across this picture from a 1967 issue of American Rifleman.  It's a machine rest at the Lake City Army Ammunition Plant used in accuracy testing of their "National Match" ammunition.

I shot a 1 shot group yesterday that was 0 MOA.

Originally Posted By Molon: Good eye!  The extreme spread of your group measures 2.27”.  The mean radius is 0.67”.   If you can, the next time you go shooting, fire three 10-shot groups of your XM193 from 100 yards and post your results.  I can then make more of an “apples to apples” comparison.

I will do exactly that.  Thanks for running it through your software!

What lot number is the XM193 you used for this group?  Some lots definitely seem to be more accurate than others.  The limiting factors to the accuracy of XM193 are obviously a.) the bullet, but also I believe b.) the velocity to which the round is loaded.  In my personal experience I have found that 55 grain FMJ bullets fired from an AR-15 tend to be the most accurate at velocities that are downloaded about 200 fps from M193 velocities.  (Not that 55 grain FMJ bullets are known for stellar accuracy to begin with.)*

I'm embarassed to say that I don't know.  

I had a bunch of loose 20-round boxes of XM193 from several different cases (that may have been different lots) that I loaded into mags and then threw away the empty boxes and just grabbed full mags whenever I go to the range.  I just looked around the house, and there isn't a single one of the boxes left anywhere, so I cannot figure out what lot it was.

Below is a pic of the best 10-shot group from 100 yards that I have obtained using handloaded Hornady 55 grain FMJ bullets.  The extreme spread is 1.38” and the mean radius is 0.41”.  I don’t know if 55 grain FMJ bullets are capable of anything more accurate than this. home.comcast.net/~gocartmozart/hornady_55_fmj.jpg *I love this quote from the book Black Magic: For accuracy usage, the 55 grain FMJ is an evil bullet... Under sufficient magnification, it wouldn't surprise me a bit to find a small "666" tattooed into its shiny little ogival head. Abandon all hope, ye who seek tiny groups here!

Even with my relatively lack of experience in centrefire precision shooting, I can tell an improvement of 68gr, 69 gr and 75 gr over 55 gr.

Just for giggles, here's the best 55 gr. 10-shot group that I've pulled off - with Black Hills 55gr blue box



The picture is cropped a little too tight to see the 1-inch squares, but I think a quarter is right around an inch (or 15/16 of an inch) - so the group probably has a spread of around 1 2/3"

Originally Posted By Molon: Originally Posted By GHPorter: I want to know more about that machine rest.  How about more pictures, a web site, anything! I've finished going through all my back issues of Shooting Times and I haven't come across any more pictures of Rick Jamison's machine rest.  However, I did just recently come across this picture from a 1967 issue of American Rifleman.  It's a machine rest at the Lake City Army Ammunition Plant used in accuracy testing of their "National Match" ammunition. home.comcast.net/~gocartmozart/macine_rest_1967.jpg

That one makes loads of sense: a barreled action (a 1903, I think) in a well-built clamping structure with a shield over the action in case "anything bad" happens.  This "clamped, barreled action" form is pretty similar to the "I-beam" unlimited benchrest rifles I've seen mentioned (Dean Grinell's ABC's of Reloading from many years ago had an article on one such competitor who made every one of his bullets by hand, and his "rifle" was a heavy barreled action attached to a hefty piece of I-beam...with a scope, too).  Now I just have to figure out how to convince some smith around here that I'm not on something when I start asking about "some sort of clamping device I can use to make a barreled action into a machine rest rifle"  I'll post responses if they're funny enough!  Thanks for all the research, Molon!

Originally Posted By DK-Prof: Just for giggles, here's the best 55 gr. 10-shot group that I've pulled off - with Black Hills 55gr blue box photos.ar15.com/ImageGallery/Attachments/DownloadAttach.asp?iImageUnq=47084

Your group makes for a very interesting comparison.  The extreme spread measures 1.33" and more importantly the mean radius measures 0.43".  



Look how closely these measurements compare to those from my best 10-shot group of handloaded Hornady 55 grain FMJ rounds with an extreme spread of 1.38" and a mean radius of 0.41".



The mean radii of these two groups differ by only 2/100ths of an inch.  I think we are looking at the accuracy threshold for a single 10-shot group of the 55 grain FMJ bullet.  This indicates the accuracy limitations of the 55 grain FMJ bullet and that the 55 grain FMJ pill is just not capable of minute of angle accuracy, (even though our hero the Internet Commando would have us believe otherwise.)

I also think your group is a testament to the quality of Black Hills ammunition.  Their remanufactured ammunition (at least their 55 grain FMJ load) rivals the accuracy of  carefully assembled 55 grain FMJ handloads.

Here is some additional information on U.S. M193 from a 1988 back-issue of Rifle magazine.

The accuracy requirement from a test fixture calls for a maximum of a two-inch mean radius at 200 yards from ten 10-shot groups (which equates to approximately three MOA).

Here is an excerpt from a 1974 issue discussing the benefits of using the mean radius for measuring groups.

These examples illustrate the sensitiveness of the extreme spread to number of shots in the group.  Indeed, as the table indicates, the measures made to only the outside shots of the group, e.g. the extreme spread, are very sensitive to number of shots, while the measures made to all the shots, e.g. the mean radius are far less so.  It may be added that the latter measures are also less variable in their representation of the group; they are more efficient.  This explains why the target testing of U.S. military rifle ammunition is by mean radius.

Outstanding information.

Thanks!

Hey DK-Prof,

Where are those three 10-shot groups of XM193?

Molon

Here is another example of using the mean radius to evaluate the accuracy of a rifle/ammunition combination.  I am currently working on a comparison of Hornady’s 75 grain TAP FPD (For Personal Defense) load and their 75 grain 5.56 TAP T2 load for a thread in the ammunition forum.  The preliminary results of accuracy testing are quite pertinent to this thread also.  

I fired three 10-shot groups of the 75 grain TAP FPD from 100 yards using a Colt 20” HBAR that is chrome lined, NATO chambered and has a 1:7” twist.  I then overlayed those groups on each other using the RSI Shooting Lab software program for a 30-round composite target, (the target on the left below).  The target on the right is the same 30-round composite target of XM193 used previously in this thread, (which was fired from a Colt 16” HBAR).  Once again the visual comparison makes it obvious which is the more consistent ammunition/rifle combination.

The mean radius of the XM193 composite target is .88” while the mean radius of the TAP FPD target is .41”.  Using the mean radius method makes it easy to objectively and quantifiably show the improvement in accuracy by using the TAP ammunition over the XM193 as used in this example.

Originally Posted By Molon: Hey DK-Prof, Were are those three 10-shot groups of XM193? Molon

I'm sorry that I'm not quite sure which three groups you're referring to

But I rummaged through some relatively recent pictures, and I'm sure the following groups are all XM193 (the first one is the 17-round group that you already looked at in this thread - but I'm not 100% sure if you've already examined the next three groups, since I may have posted them in another thread a while ago).

Originally Posted By DK-Prof: Originally Posted By Molon: Good eye!  The extreme spread of your group measures 2.27”.  The mean radius is 0.67”.   If you can, the next time you go shooting, fire three 10-shot groups of your XM193 from 100 yards and post your results.  I can then make more of an “apples to apples” comparison. I will do exactly that.  Thanks for running it through your software!

That's a good group for XM193.  Nice shooting!

I started out doing 120-shot groups because the objective was just to blast away and break-in the rifles. With my 24" free floated Ar-15, I made a 2" hole at 25 yd using Wolf 55/62 FMJ/JHP (whatever I had in the mags). That's approximately an 8" hole at 100 yd.

I shoot one shot groups.

Originally Posted By DK-Prof: Originally Posted By Molon: Hey DK-Prof, Were are those three 10-shot groups of XM193? Molon I'm sorry that I'm not quite sure which three groups you're referring to

Never mind - I'm stupid and got confused about what you meant, because you accidentally left out an "h" in "where", so instead of realizing you were asking about where my NEW groups are, I mistakenly thought you were asking if groups I posted previously were xm193.  Sorry about the misunderstanding.  In retrospect, it's completely obvious that it was just a typo.  


Fortunately, I DID go shoot today, and remembered to shoot three brand new 10-shot groups of XM193.

It was really nice outside as well.  About 75 degrees, a very slight breeze, but very humid.




...btw, the hole in the right upper hand corner of the last group is not a shot, but the hole from the roofing nails that hold the target to the stand.)


Since we earlier also discussed how accurate the 55gr bullet is even capable of being, I tried some ammo I had never tried before - "UltraMax" 55gr soft point and Federal Premium Gold 55 gr nosler ballistic tip.  The best I did with two groups of each was this group, which looks like 1 MOA (but the second group was not quite as good).  This was a group of the UltraMax, which was a LOT cheaper than the Federal Gold.

Originally Posted By DK-Prof: Originally Posted By DK-Prof: Originally Posted By Molon: Hey DK-Prof, Were are those three 10-shot groups of XM193? Molon I'm sorry that I'm not quite sure which three groups you're referring to Never mind - I'm stupid and got confused about what you meant, because you accidentally left out an "h" in "where", so instead of realizing you were asking about where my NEW groups are, I mistakenly thought you were asking if groups I posted previously were xm193.  Sorry about the misunderstanding.  In retrospect, it's completely obvious that it was just a typo.   Fortunately, I DID go shoot today, and remembered to shoot three brand new 10-shot groups of XM193. It was really nice outside as well.  About 75 degrees, a very slight breeze, but very humid. photos.ar15.com/ImageGallery/Attachments/DownloadAttach.asp?iImageUnq=47376 photos.ar15.com/ImageGallery/Attachments/DownloadAttach.asp?iImageUnq=47377 photos.ar15.com/ImageGallery/Attachments/DownloadAttach.asp?iImageUnq=47378 ...btw, the hole in the right upper hand corner of the last group is not a shot, but the hole from the roofing nails that hold the target to the stand.) Since we earlier also discussed how accurate the 55gr bullet is even capable of being, I tried some ammo I had never tried before - "UltraMax" 55gr soft point and Federal Premium Gold 55 gr nosler ballistic tip.  The best I did with two groups of each was this group, which looks like 1 MOA (but the second group was not quite as good).  This was a group of the UltraMax, which was a LOT cheaper than the Federal Gold. photos.ar15.com/ImageGallery/Attachments/DownloadAttach.asp?iImageUnq=47379

I guess I should actually read what I post, my fault entirely.  The "H" has been added to my above post.

Now, as for your new groups from today, that is exactly what I was looking for.  The three individual groups measure 1.98", 2.57" and 2.82" for an average of 2.45".  More importantly, the mean radius of the 30-round composite target measures 0.80".  This group is much more typical of my experience with XM193.  Contrast this mean radius with the 0.41" mean radius of the composite target of the 75 grain TAP FPD load a few posts above.  The three 10-shot groups that made up that composite target measured measured 1.08", 1.15" and 1.33".



I should also clarify another point I made in my previous posts.  When I talk about the accuracy limitations of  55 grain bullets, I am speaking strictly of the full metal jacket variety.  Some the best 10-shot groups from 100 yards that I have ever fired came from Sierra 55 grain BlitzKings.

As for your 10-shot group using the 55 grain UltraMax ammo:  1.02"!

Originally Posted By Molon: Originally Posted By DK-Prof: Originally Posted By DK-Prof: Originally Posted By Molon: Hey DK-Prof, Were are those three 10-shot groups of XM193? Molon I'm sorry that I'm not quite sure which three groups you're referring to Never mind - I'm stupid and got confused about what you meant, because you accidentally left out an "h" in "where", so instead of realizing you were asking about where my NEW groups are, I mistakenly thought you were asking if groups I posted previously were xm193.  Sorry about the misunderstanding.  In retrospect, it's completely obvious that it was just a typo.   Fortunately, I DID go shoot today, and remembered to shoot three brand new 10-shot groups of XM193. It was really nice outside as well.  About 75 degrees, a very slight breeze, but very humid. photos.ar15.com/ImageGallery/Attachments/DownloadAttach.asp?iImageUnq=47376 photos.ar15.com/ImageGallery/Attachments/DownloadAttach.asp?iImageUnq=47377 photos.ar15.com/ImageGallery/Attachments/DownloadAttach.asp?iImageUnq=47378 ...btw, the hole in the right upper hand corner of the last group is not a shot, but the hole from the roofing nails that hold the target to the stand.) Since we earlier also discussed how accurate the 55gr bullet is even capable of being, I tried some ammo I had never tried before - "UltraMax" 55gr soft point and Federal Premium Gold 55 gr nosler ballistic tip.  The best I did with two groups of each was this group, which looks like 1 MOA (but the second group was not quite as good).  This was a group of the UltraMax, which was a LOT cheaper than the Federal Gold. photos.ar15.com/ImageGallery/Attachments/DownloadAttach.asp?iImageUnq=47379 I guess I should actually read what I post, my fault entirely.  The "H" has been added to my above post.

I wasn't trying to beat you up or anything, just trying to explain my previously dumb post.  

Now, as for your new groups from today, that is exactly what I was looking for.  The three individual groups measure 1.98", 2.57" and 2.82" for an average of 2.45".  More importantly, the mean radius of the 30-round composite target measures 0.80".  This group is much more typical of my experience with XM193.  Contrast this mean radius with the 0.41" mean radius of the composite target of the 75 grain TAP FPD load a few posts above.  The three 10-shot groups that made up that composite target measured measured 1.08", 1.15" and 1.33". home.comcast.net/~gocartmozart/dk_composite_02.jpg

Awesome - thanks for calculating that.

(I cannot actually see your pictures - but got the red x - but it could be a problem at my end)

I should also clarify another point I made in my previous posts.  When I talk about the accuracy limitations of  55 grain bullets, I am speaking strictly of the full metal jacket variety.  Some the best 10-shot groups from 100 yards that I have ever fired came from Sierra 55 grain BlitzKings.

Ah - that makes much more sense, and also illustrates how inexperienced I am with fancy-schmancy bullets beyond regular military FMJ.  

As for your 10-shot group using the 55 grain UltraMax ammo:  1.02"!home.comcast.net/~gocartmozart/woohoo.gif

W00t!  I am getting there!  


Thanks for all the fantastic work you have done in this thread so far, and all the help you've provided for me (in this thread and in other threads).  

Very cool info however I am a bit confused when you say

"2.34" (extreme spread) divided by 3 equals 0.78" (mean radius). Actual mean radius being 0.72"."

How is the actual mean radius being acquired? Is it by using

"To obtain the mean radius of a shot group, measure the heights of all shots above an arbitrarily chosen horizontal line. Average these measurements. The result is the height of the center of the group above the chosen line. Then in the same way get the horizontal distance of the center from some vertical line, such as for instance, the left edge of the target. These two measurements will locate the group center."

Once finding the group center measure the distance from each shot to center, then adverage all the numbers?

Thanks!

Originally Posted By Archer36: Very cool info however I am a bit confused when you say "2.34" (extreme spread) divided by 3 equals 0.78" (mean radius). Actual mean radius being 0.72"." How is the actual mean radius being acquired? Is it by using "To obtain the mean radius of a shot group, measure the heights of all shots above an arbitrarily chosen horizontal line. Average these measurements. The result is the height of the center of the group above the chosen line. Then in the same way get the horizontal distance of the center from some vertical line, such as for instance, the left edge of the target. These two measurements will locate the group center." Once finding the group center measure the distance from each shot to center, then adverage all the numbers? Thanks!

In a word: yes.  Finding the mean radius of a group is acutually very easy to do (albeit rather time consuming) once you get used to doing it.  While the written explanation by itself might sound a little confusing once you've seen it done it's very easy to understand as I'll demonstrate in my next post!

Mean Radius Demonstration

Let’s say you fired a 5-shot group from 100 yards and the resulting target looks like this.  (The X-ring measures 1.5” and the 10-ring measures 3.5”.)  



The extreme spread of the group measures 2.83”, but we want to find the mean radius (or average group radius.)  In order to find the mean radius we must first find the center of the group.  By “eye-balling”  the target most people would see that the group is centered to the left of the "X-ring" and probably a little high, but we need to find the exact location of the center of the group.

Locating the Center of the Group

The first step in finding the center of the group is to find the lowest shot of the group and draw a horizontal line through the center of that shot.  



Next, find the left-most shot of the group and draw a vertical line through the center of that shot.



Now measure the distance from the horizontal line to the other four shots of the group that are above that line.  Add those numbers together and divide by the total number of shots in the group (5).



2.50” + 1.03” + 2.01” + 1.30” = 6.84”

Divide by 5 to get 1.37”.  This number is the elevation component of the center of the group.

Next we need to find the windage component of the center of the group.  From the vertical line, measure the distance to the other four shots of the group that are to the right of the line.  Add those numbers together and again divide by the total number of shots in the group (5).



1.76” + 2.54” + 0.45” + 1.19” = 5.94”

Divide by 5 to get 1.19”  This is the windage component of the center of the group.

Finding the windage and elevation components of the center of the group is the most difficult part of this process.  Once that is done the rest of the process is a piece of cake.

Using the windage and elevation components, locate the position on the target that is 1.37” (elevation component) above the horizontal line and 1.19” (windage component) to the right of the vertical line.  This location is the center of the group!




Determining the Mean Radius

Now that we have located the position of the center of the group, the first step in determining the mean radius is to measure the distance from the center of the group to the center of one of the shots.  This line is a single radius.




Now measure the distance from the center of the group to the center of each of the rest of the shots in the group.  Add the measurements of all the radii together and then divide by the total number of shots in the group (5).



0.85” + 1.35” + 1.38” + 0.84” + 1.61” = 6.03”

Divide by 5 to get 1.21”.   This is the mean radius (or average group radius) of the group!

Using the mean radius measurement to scribe a circle around the center of the group gives you a graphic representation of the mean radius.  This shows the average accuracy of all the shots in the group. This demonstrates why the mean radius is much more useful than the extreme spread in evaluating the accuracy of our rifles and ammunition.

Wow thanks for the graphic demonstration really helps a lot... going to give this a shot on my next shoot.

Keep in mind I don't do this by hand anymore (too time consuming.)  There are software programs out now that enable you to determine the mean radius of a target in less than 30 seconds.

Originally Posted By Molon: Keep in mind I don't do this by hand anymore (too time consuming.)  There are software programs out now that enable you to determine the mean radius of a target in less than 30 seconds.

Molon,
Can you tell us what this software is or give a link please?

TIA

Seydou

Here's a link to a whole list of ballistic programs.

http://www.stevespages.com/page8b.htm

hot-link

Thanks.

Seydou

Don't mention it.  Of the many programs listed in the link, two that I like to use for determining the mean radius are Group Size Calculator and RSI Shooting Lab.  

Group Size Calculator is very inexpensive ($10.50) but does require a scanner to input your targets.  RSI Shooting Lab is much more expensive ($99.95) but does not require a scanner.  Both programs require careful calibration of the program to your scanner/monitor to provide accurate results.

There are other programs that don't require calibration, but you have to measure the coordinates of the shots on your target and enter the numeric values into the program by hand.  (More time consuming.)

I've scanned the thread and didn't see this addressed; sorry if I missed it.

The mean radius has the advantage of being more easily calculated, but other than that, why is it better than a standard deviation?  If the shot distribution is roughly gaussian, the standard deviation allows description of the accuracy level for any confidence level.  I.e, 67% CL for 1 sigma, 95% CL for 2 sigma, etc.

Mean radius:

<X - <X>>

Standard deviation

sqrt(<X**2> - <X>**2)

Originally Posted By bnorman: I've scanned the thread and didn't see this addressed; sorry if I missed it. The mean radius has the advantage of being more easily calculated, but other than that, why is it better than a standard deviation?  If the shot distribution is roughly gaussian, the standard deviation allows description of the accuracy level for any confidence level.  I.e, 67% CL for 1 sigma, 95% CL for 2 sigma, etc. Mean radius: <X - <X>> Standard deviation sqrt(<X**2> - <X>**2)

Standard deviation from what?

I don't think it's an either-or question, but rather a matter of whether you want to ADD the more rich information of standard deviation to a simple report of mean radius.  I agree completely that standard deviation would provide additional valuable information.

The way I see it, there are increasingly accurate ways of reporting a clustering of holes in a piece of paper (assuming 10-shot groups).

1. The least informative is extreme spread, which is what most people use.
2. Better is mean radius, as Molon has explained.
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).

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.

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.

Got it (sorry I misunderstood) - that WOULD be even better, you are right.

Molon,

I think this would be a great opportunity to use PowerPoint to develop training slides

Um moderators, why in the heck hasn't this been tacked yet? This is a no BS informative thread. It is threads like these that motivate me to send money every year.

tag

Good question!  First let me point out that nowhere in this thread has anyone ever stated that the use of the mean radius was better  than using the standard deviation.  One of the main points of this thread is that using the mean radius method is far superior to using the extreme spread to measure groups.  

The mean radius method is the convention used by the U.S. military in testing the accuracy of ammunition.  I started using it after reading about it in Hatcher’s Notebook, the NRA Firearms Fact Book, and back issues of American Rifleman.