Posted: 4/24/2006 11:06:41 PM EDT [Last Edit: 4/23/2020 10:49:52 AM EDT by Molon] The Trouble With 3-Shot Groups



(The Distant Prequel to The Trouble With Tribbles)





As our story begins, we find our Hero sitting down at the shooting bench on the 100 yard line at his local range. Our Hero is a real Internet Commando. He has watched the movie Heat 87 times, is registered on all of the important Internet firearms forums, knows the difference between a "clip” and a "magazine” and attends every local gun show without fail.



His latest acquisition from the local gun show is a 16” barreled Frankengun that originally belonged to a Delta Force operator who was at the Battle of Mogadishu. (He could tell you the name of the operator, but then he’d have to k……well, you know.) The seller at the gun show let our Hero in on another secret too. Eugene Stoner himself helped the Delta operator assemble this particular AR-15.



Our hero settles into position on the bench and fires his first 3-shot group. (He only fires 3-shot groups because everyone from the Internet knows that you only have to fire 3-shot groups to test the accuracy of your rifle and ammunition.)



His first group forms a nice little triangle at the bottom of the 10-ring. Our Hero just knew this was going to be a good day. After all, he was using that XM193 ammunition. A former SEAL who was selling beef jerky at the gun show had told our Hero that the SEALs used ammo just like XM193 in Viet Nam and that a single shot in the buttocks from this ammo would "blow your head clean off!”



At the cease fire our Hero checks his first target. (Our Hero’s targets are NRA High Power type targets reduced for 100 yards. The X-ring measures 1.5” and the 10-ring measures 3.5”). Much to his satisfaction, his first group measures 1.16”, almost minute of angle! Now most Internet Commando wannabees would have stopped right there and gone home and spent the next 3 hours posting threads on the internet about their great accomplishment, but not our Hero. He makes the decision….to fire a second group!



group one



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Our Hero has read reports of other Internet Commandos who have been able to achieve sub-minute of angle groups using XM193. After all, this ammo uses "Full Metal Jacket” bullets and is made to "Mil-spec.”



Our Hero carefully watches the bench-rest shooter three lanes down (who monitors the wind conditions on the range using a Wind Probe) and only fires immediately after the bench-rester does. He does this for three shots and then checks his target at the next cease fire.



Our Hero measures this target three times just to be sure. The group measures 0.93”! Hooah, a sub-minute of angle group! Our Hero is now one of the few, the proud, the real Internet Commandos who can claim to shoot sub-minute of angle groups using XM193!



group two



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There’s just one little problem with that second group our Hero fired. It is centered in a different location on the target than the first one. The second group is centered 0.12” above the center of the target and the first group is centered 1.44” below the center of the target. Hmmm…



Well our Hero is a real Internet Commando so he can’t let details bother him. After all, sub-minute of angle is sub-minute of angle!



As our Hero starts to pack up his targets, something in the back of his mind starts to nag at him. He recalls the reports of the other Internet Commandos. They didn’t just shoot sub-minute of angle groups with XM193; they did it "all day long.”



Well, not wishing to be looked down upon by the other Internet Commandos our Hero decides to shoot one more 3-shoot group. After all, if his carbine and ammo could shoot two, sub-minute of angle groups, they could surely do it "all day long.”



So, our Hero settles back into position and again taking his cue from the bench-rester fires a third 3-shot group. Upon retrieving his third target and measuring the group our Hero can not believe his eyes. The group measures 2.5”! How can this be possible? He was using XM193 and not just any XM193. It was the fabled LOT #3, the most accurate and hard to come by of all the lots, yet this third group was larger than the first two groups combined! Hmmm… Our Hero wonders how he can ever show his avatar on an Internet forum again after firing such a group with XM193.



group three



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Then, slowly our Hero starts to recall a word he has heard mentioned many times before on his favorite Internet forum. It starts with the letter F. Hmm . . .F . . .Fl . . . Flyer! That’s it, flyer! That low shot down at seven o’clock on the target is a flyer! It’s not the fault of the gun or the ammo, it’s a flyer. It's caused by user error, the loose nut behind the stock, the wind, the sun or any other excuse that can be dreamt of in your philosophy, but not the rifle or ammo.



The flyer is something to be discounted as if it never happened. (Why be concerned with reality when you are an Internet Commando?) Since that shot is discounted, why not discount that whole group as if it never happened? (After all, isn’t that what an Internet Commando does?) Our Hero decides to discount the entire group and throws the target in the trash. He makes a solemn vow to never mention this group to anyone. After all, he is a real Internet Commando.



Upon returning home our Hero makes all the usual posts on the Internet about his sub-minute of angle groups using XM193. True to his solemn vow, he makes no mention of his 2.5” group. At no time does he mention that these groups were 3-shot groups. Nor does he make any mention of the fact that the groups were centered in different locations on the target.



Our Hero ends his day wondering if his grandson, or great-grandson or even great great-grandson will remember the accomplishments of the real Internet Commandos or if their contributions to the shooting world will eventually be lost in time? He even wonders what his progeny might someday be named, James or possibly Tiberius? Hmmm…





While the fable above is obviously fiction, the examples of targets shown are based on a real target of a 10-shot group fired from 100 yards using XM193. When all of the 3-shot groups from above are overlayed on a single target you see a much truer example of the limitations of the rifle and ammo. The gap made by the so-called “flyer” is filled in by the other seven shots of the whole group. You can see that when the first 3-shot group (which is centered 1.5” below the second 3-shot group) is displayed with the second 3-shot group you are actually seeing a much better indication of the total dispersion of the rifle and ammo combination. Most people fail to mention that their 3-shot groups are impacting at different locations on the target. This is why 10-shot groups are a much better indicator of a the radial dispersion of a rifle/ammunition combination.





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the real target



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Rick Jamison, the author of the Precision Reloading column in Shooting Times magazine approaches accuracy testing in a scientific manner. He uses a machine rest for testing and fires 10-shot groups. Here are his own words on the subject from one of his articles:



"There are stories of a single bullet that for no explained reason flies out of what might have been a tight cluster. This often occurs with a three-shot string and many times with a five-shot string. If you're lucky enough to fire a group without a flier, you can end up with a very tight group. However, usually what happens if another five or seven shots are fired to complete a 10-shot string, other bullets fill in the space between the main group and the flier to make a reasonably rounded group. Ten shots are a more reliable indicator when it comes to predicting what a load is likely to do in the future.



The problem with 10-shot groups is that when you report them, everyone thinks you aren't shooting very well or that the ammunition is not good because the group sizes are so much larger than three- or five-shot groups. Also, when we're firing three- or five-shot groups with a flier, it is only natural to assume that it was caused by a flinch or “pulling” the shot. Therefore, since the flier was our own fault, the tendency is to eliminate it from any reporting of group size."





Here is another quote about 3-shot groups from another Precision Reloadingarticle by Rick Jamison."Some shooters may have two or three three-shot groups to prove the load is really accurate. It really takes more shooting than that to make a judgment on a load’s accuracy potential. Three shots forming a tight cluster is nice to look at, but it is little more than an accident. Shooting three-shot groups to see how everything is working is essentially a waste of time and components."







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Here’s another example demonstrating that 3-shot groups do not provide a valid indicator of the radial dispersion of a rifle/ammunition combination. The test vehicle for this demonstration was a one of my AR-15s that has a free-floated Colt HBAR and is wearing a Leupold 3.5-10X40 LR/T that has ¼ MOA clicks for windage and elevation.







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Prior to beginning this demonstration, I fired a 10-shot control group at a distance of 50 yards using a hand-load topped with the Sierra 77 grain MatchKing. The target used is an NRA High Power type target reduced for 50 yards. The X-ring measures 0.75" and the 10-ring measures 1.75". The 10-shot control group had an extreme spread of 0.493”.





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Next, I fired a couple of 3-shot groups from 50 yards using Federal XM193 ammunition. (Shooting was conducted at an indoor range, so wind was not a variable in this case.) The two targets are pictured below.



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The center of the first 3-shot group was located approximately ¼ MOA high and ½ MOA to the right of the center of the target, which was the point-of-aim. However, the center of the second 3-shot group was located approximately 1.5 MOA low and ½ MOA to the left of the center of the target. The centers of just these two, 3-shot groups are almost 2 MOA apart in elevation.







The centers of the individual 3-shot groups are indicated below by the blue crosses.



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I actually fired a total of ten, 3-shot groups in a row with the XM193 ammunition for a total of thirty shots. Thirty occurrences of something that is being studied is what the statistics types like to refer to as a "large sample." Using the RSI Shooting Lab software program I was able to overlay all ten of the 3-shot groups on each other for a 30-shot composite group.





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This 30-shot composite group gives us a much better picture of the radial dispersion of the rifle/ammunition combination (and in this case particularly the ammunition) than 3-shot groups by themselves do. These 30 shots also give us give us a much better indication of where the center of the points-of-impact are.







The ten, 3-shot groups that were fired in a row for the above demonstration measured (from smallest to largest):



0.40”

0.57”

0.64”

0.74”

0.80”

0.85”

1.03”

1.13”

1.19”

1.54”



The variation between the smallest and largest group above is 74%!



In addition to firing the ten, 3-shot groups listed above, I also fired three, 10-shot groups in a row from 50 yards for another thirty rounds of XM193.



The groups measured (from smallest to largest):



1.24”

1.46”

1.97”



The variation between the smallest and largest of these three groups is only 37%. Using the same total number of rounds in a set, but firing one set using ten, 3-shot groups and the other using three, 10-shot groups shows that the group to group variation of the 10-shot groups is far less than that of the 3-shot groups, half as much in this case. This shows again that 10-shot groups are a more consistent indicator of the radial dispersion of a rifle/ammunition combination or as Rick Jamison stated, “Ten shots are a more reliable indicator when it comes to predicting what a load is likely to do in the future.”







Before I can continue with The Trouble With 3-Shot Groups, I need to define the concept of “mean radius” (also called average group radius.)





A Primer on the Mean Radius





The mean radius is a method of measurement of the radial dispersion of shot-groups that takes into account every shot in the group. It provides a more useful analysis of the consistency of ammunition and firearms than the commonly used method of extreme spread.



Mean radius as defined in Hatcher's Notebook “is the average distance of all the shots from the center of the group. It is usually about one third the group diameter (extreme spread)” for 10-shot groups. (The ratio is actually closer to 3.2 times the mean radius = the extreme spread, for 10-shot groups, depending on the sample size and the morphology of the particular groups sampled.)



To obtain the mean radius of a shot-group, measure the heights of all shots above the lowest shot in the group. Average these measurements. The result is the height of the center of the group above the lowest shot. Then in the same way, get the horizontal distance of the center from the shot farthest to the left. These two measurements will locate the group center. Now measure the distance of each shot from this center. The average of these measures is the mean radius.





Once you get the hang of measuring groups using the mean radius it becomes very simple to do. While being very simple to do, it is also very time consuming. Modern software programs such as RSI Shooting Lab and On Target make determining the mean radius a snap.



The picture below is a screen capture from RSI Shooting Lab. The red cross is the center of the group (a little high and right of the aiming point). The long red line shows the two shots forming the extreme spread or group size. The yellow line from the red cross to one of the shots is a radius. Measure all the radii and take the average to obtain the mean radius.





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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”.)





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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.





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







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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).







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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).









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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!







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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”.





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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).



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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 radial dispersion of our rifles and ammunition.







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Here are some interesting quotes from old issues of American Rifleman on the subject:



“Mean radius is the mean distance of bullet impacts from center of the test group. It is used in government ammunition acceptance because it takes account of every shot and comes close to maximizing the test information. While there is no exact relationship between this measure and the simpler and more convenient group diameter, the 10-shot group diameter averages slightly over 3 times the mean radius.”



"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."



.....





Over-laying the targets of at least three, 10-shot groups fired in a row and determining the mean radius of the composite group gives us a statistically powerful tool for evaluating the radial dispersion of a rifle/ammunition combination.

















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As explained previously, the mean radius method of measuring groups gives us information about each shot in the group, not just the two worst shots of the group as is the case when measuring the extreme spread. The following comparisons will be made using the mean radius of the different groups.



Besides doing a 30-shot composite group formed from the ten, 3-shot groups fired in a row with XM193 as shown above, I also compiled a 30-shot composite group formed from the three, 10-shot groups fired in row. Here they are, side by side for comparison.







The 30-shot composite group formed from the ten, 3-shot groups has a mean radius of 0.45” (indicated by the inner blue circle.)



The 30-shot composite group formed from the three,10-shot groups has a mean radius of 0.42” (also indicated by the inner blue circle.)



Finally, I compiled a 60-shot composite group formed from all of the above groups.















The mean radius of the 60-shot composite group is 0.44”. This sixty-shot composite group gives us a fairly definitive idea of what we can expect from the rifle/ammunition combination in question when fired from 50 yards. Just as importantly, this demonstrates that the thirty-shot composite groups come quite close to the same results as the 60-shot composite group; differing by only a couple one-hundredths of an inch in mean radius from the 60-shot composite group.







....





Since the mean radius method of measuring groups gives us a better picture of what is occuring with all the shots in the group, we can use the mean radius measurements to get more accurate comparisons between groups. The mean radii of the ten, 3-shot groups from above are as follows (from smallest to largest):



0.17”

0.23”

0.25”

0.30”

0.32”

0.32”

0.46”

0.50”

0.50”

0.69”



The 3-shot groups have a variation of 75% from smallest to largest!



The mean radii of the three, ten-shot groups from above measure (from smallest to largest):



0.34”

0.39”

0.50”



The mean radii of the 10-shot groups only have a group to group variation of 32%, less than half that of the 3-shot groups. Once again this shows the vastly improved consistency of 10-shot groups compared to 3-shot groups.



If you recall from the above post, the sixty-shot composite group had a mean radius of .44”. Look at how much closer the mean radii of the individual 10-shot groups come to the mean radius of the sixty-shot composite group than the mean radii of the 3-shot groups do. The mean radius of the 30-shot composite group formed from over-laying the three 10-shot groups on each other is even closer. Consistency and a high degree of predictability are what make the use of 10-shot groups (particularly three,10-shot groups fired in a row) far superior to 3-shot groups in evaluating the precision of our rifles and ammunition!







....





Here is a final demonstration that further illustrates and reinforces all the concepts previously presented. Using one of my AR-15s that has a free-floated Colt HBAR, I fired five, 10-shot groups from 100 yards using a hand-load topped with 55 grain FMJ bullets. The five 10-shot groups were over-layed on each other to produce a 50-round composite group with a mean radius of 0.70”. The average extreme spread for all five groups was 2.30”.















I then proceeded to fire another 50 rounds of the same hand-load in 3-shot groups, for a total of sixteen, 3-shot groups (with an extra 2-shot group at the end.) Several of the groups were sub-MOA.



group #06: 0.66”

group #12: 0.92”

group #13: 0.59”

group #15: 0.41”

group #16: 0.63”



However, several other groups had extreme spreads of well over 2”.



group #02: 2.53”

group #07: 2.89”

group #08: 2.39”

group #09: 2.86”

group #11: 2.47”



Those who espouse the use of 3-shot groups for evaluating the radial dispersion of a rifle/ammunition combination like to point to the smallest 3-shot groups for their accuracy claims and then pretend that all those large 3-shot groups don’t exist.



The average extreme spread for all sixteen of the 3-shot groups was 1.72”. Also, the statistical centers of many of the groups, (both large and small) were located at different locations on the target.



Pictured below is the first 3-shot group of the 55 grain FMJ load and beneath that the first three, 3-shot groups over-layed on each other.













As we continue to sequentially over-lay the 3-shot groups on each, the pieces of the puzzle begin to fill in the gaps giving us a more complete view of the entire picture.

















Finally, when all of the 50 rounds fired in 3-shot groups have been over-layed on each other, we have a composite group that is remarkably similar in radial dispersion to the 50 round composite group formed from the five 10-shot groups.













Here’s the only target from the above shooting session that the Internet Commando would have posted.









Remember, all the data for the above examples were obtained during live-fire testing from my





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Institutions and organizations that buy enormous amounts of ammunition and weapons are far more interested in facts, than sales hype and propaganda and most of them demand that accuracy/precision testing be conducted using 10-shot groups. This includes the Federal Bureau of Investigation, Crane Naval Surface Warfare Center, the US Army Marksmanship Unit and the US military’s acceptance testing of both 5.56mm ammunition and weapons. On the other hand, businesses that use 3-shot groups for making their accuracy claims are usually trying to sell something.



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At the 1992 Barcelona Olympics, USA Shooting Team members Launi Meili and Robert Foth won the gold and silver medals in the three-position rifle events. The Olympians used the new Federal Gold Medal ammunition to aid them in obtaining their victories. This was the first time in more than 30 years that an American won an Olympic medal in one of the small bore shooting events while using American-made ammunition.



It’s interesting to note that pertaining to the accuracy/precision development and multifaceted testing of the Federal ammunition that helped the US Olympians win gold and silver medals in Barcelona, Federal’s Director of Product Engineering, Dave Longren, had this to say:



'The standard test string was three 10-shot groups, with the most attention paid to the 30-shot composite. "When you’re working at this level, the traditional five 5-shot group test simply doesn’t give you statistically valid results.”'*



* Hunnicutt, Robert. “Ammo Good as Gold.” American Rifleman Nov. 1992: 32-33, 72-73. Print.



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"We all use math every day. To predict weather, to tell time, to handle money -- math is more than formulas and equations; it's logic, it's rationality. It's using your mind to solve the greatest mysteries we know."



Charlie Eppes from NUMB3RS





. . . and it helps us evaluate and improve our rifle-craft.



Molon







......... (The Distant Prequel toAs our story begins, we find our Hero sitting down at the shooting bench on the 100 yard line at his local range. Our Hero is a real Internet Commando. He has watched the movie87 times, is registered on all of the important Internet firearms forums, knows the difference between a "clip” and a "magazine” and attends every local gun show without fail.His latest acquisition from the local gun show is a 16” barreled Frankengun that originally belonged to a Delta Force operator who was at the Battle of Mogadishu. (He could tell you the name of the operator, but then he’d have to k……well, you know.) The seller at the gun show let our Hero in on another secret too. Eugene Stoner himself helped the Delta operator assemble this particular AR-15.Our hero settles into position on the bench and fires his first 3-shot group. (He only fires 3-shot groups because everyone from the Internet knows that you only have to fire 3-shot groups to test the accuracy of your rifle and ammunition.)His first group forms a nice little triangle at the bottom of the 10-ring. Our Hero just knew this was going to be a good day. After all, he was using that XM193 ammunition. A former SEAL who was selling beef jerky at the gun show had told our Hero that the SEALs used ammo just like XM193 in Viet Nam and that a single shot in the buttocks from this ammo would "blow your head clean off!”At the cease fire our Hero checks his first target. (Our Hero’s targets are NRA High Power type targets reduced for 100 yards. The X-ring measures 1.5” and the 10-ring measures 3.5”). Much to his satisfaction, his first group measures 1.16”, almost minute of angle! Now most Internet Commando wannabees would have stopped right there and gone home and spent the next 3 hours posting threads on the internet about their great accomplishment, but not our Hero. He makes the decision….to fire a second group!group oneOur Hero has read reports of other Internet Commandos who have been able to achieve sub-minute of angle groups using XM193. After all, this ammo uses "Full Metal Jacket” bullets and is made to "Mil-spec.”Our Hero carefully watches the bench-rest shooter three lanes down (who monitors the wind conditions on the range using a Wind Probe) and only fires immediately after the bench-rester does. He does this for three shots and then checks his target at the next cease fire.Our Hero measures this target three times just to be sure. The group measures 0.93”! Hooah, a sub-minute of angle group! Our Hero is now one of the few, the proud, the real Internet Commandos who can claim to shoot sub-minute of angle groups using XM193!group twoThere’s just one little problem with that second group our Hero fired. It is centered in a different location on the target than the first one. The second group is centered 0.12” above the center of the target and the first group is centered 1.44” below the center of the target. Hmmm…Well our Hero is a real Internet Commando so he can’t let details bother him. After all, sub-minute of angle is sub-minute of angle!As our Hero starts to pack up his targets, something in the back of his mind starts to nag at him. He recalls the reports of the other Internet Commandos. They didn’t just shoot sub-minute of angle groups with XM193; they did it "all day long.”Well, not wishing to be looked down upon by the other Internet Commandos our Hero decides to shoot one more 3-shoot group. After all, if his carbine and ammo could shoot two, sub-minute of angle groups, they could surely do it "all day long.”So, our Hero settles back into position and again taking his cue from the bench-rester fires a third 3-shot group. Upon retrieving his third target and measuring the group our Hero can not believe his eyes. The group measures 2.5”! How can this be possible? He was using XM193 and not just any XM193. It was the fabled LOT #3, the most accurate and hard to come by of all the lots, yet this third group was larger than the first two groups combined! Hmmm… Our Hero wonders how he can ever show his avatar on an Internet forum again after firing such a group with XM193.group threeThen, slowly our Hero starts to recall a word he has heard mentioned many times before on his favorite Internet forum. It starts with the letter F. Hmm . . .F . . .Fl . . . Flyer! That’s it, flyer! That low shot down at seven o’clock on the target is a flyer! It’s not the fault of the gun or the ammo, it’s a flyer. It's caused by user error, the loose nut behind the stock, the wind, the sun or any other excuse that can be dreamt of in your philosophy, but not the rifle or ammo.The flyer is something to be discounted as if it never happened. (Why be concerned with reality when you are an Internet Commando?) Since that shot is discounted, why not discount that whole group as if it never happened? (After all, isn’t that what an Internet Commando does?) Our Hero decides to discount the entire group and throws the target in the trash. He makes a solemn vow to never mention this group to anyone. After all, he is a real Internet Commando.Upon returning home our Hero makes all the usual posts on the Internet about his sub-minute of angle groups using XM193. True to his solemn vow, he makes no mention of his 2.5” group. At no time does he mention that these groups were 3-shot groups. Nor does he make any mention of the fact that the groups were centered in different locations on the target.Our Hero ends his day wondering if his grandson, or great-grandson or even great great-grandson will remember the accomplishments of the real Internet Commandos or if their contributions to the shooting world will eventually be lost in time? He even wonders what his progeny might someday be named, James or possibly Tiberius? Hmmm…While the fable above is obviously fiction, the examples of targets shown are based on a real target of a 10-shot group fired from 100 yards using XM193. When all of the 3-shot groups from above are overlayed on a single target you see a much truer example of the limitations of the rifle and ammo. The gap made by the so-called “flyer” is filled in by the other seven shots of the whole group. You can see that when the first 3-shot group (which is centered 1.5” below the second 3-shot group) is displayed with the second 3-shot group you are actually seeing a much better indication of the total dispersion of the rifle and ammo combination. Most people fail to mention that their 3-shot groups are impacting at different locations on the target. This is why 10-shot groups are a much better indicator of a the radial dispersion of a rifle/ammunition combination.the real targetRick Jamison, the author of the Precision Reloading column in Shooting Times magazine approaches accuracy testing in a scientific manner. He uses a machine rest for testing and fires 10-shot groups. Here are his own words on the subject from one of his articles:"There are stories of a single bullet that for no explained reason flies out of what might have been a tight cluster. This often occurs with a three-shot string and many times with a five-shot string. If you're lucky enough to fire a group without a flier, you can end up with a very tight group. However, usually what happens if another five or seven shots are fired to complete a 10-shot string, other bullets fill in the space between the main group and the flier to make a reasonably rounded group.The problem with 10-shot groups is that when you report them, everyone thinks you aren't shooting very well or that the ammunition is not good because the group sizes are so much larger than three- or five-shot groups. Also, when we're firing three- or five-shot groups with a flier, it is only natural to assume that it was caused by a flinch or “pulling” the shot. Therefore, since the flier was our own fault, the tendency is to eliminate it from any reporting of group size."Here is another quote about 3-shot groups from another Precision Reloadingarticle by Rick Jamison."Some shooters may have two or three three-shot groups to prove the load is really accurate. It really takes more shooting than that to make a judgment on a load’s accuracy potential. Three shots forming a tight cluster is nice to look at, but it is little more than an accident. Shooting three-shot groups to see how everything is working is essentially a waste of time and components."....Here’s another example demonstrating that 3-shot groups do not provide a valid indicator of the radial dispersion of a rifle/ammunition combination. The test vehicle for this demonstration was a one of my AR-15s that has a free-floated Colt HBAR and is wearing a Leupold 3.5-10X40 LR/T that has ¼ MOA clicks for windage and elevation.Prior to beginning this demonstration, I fired a 10-shot control group at a distance of 50 yards using a hand-load topped with the Sierra 77 grain MatchKing. The target used is an NRA High Power type target reduced for 50 yards. The X-ring measures 0.75" and the 10-ring measures 1.75". The 10-shot control group had an extreme spread of 0.493”.Next, I fired a couple of 3-shot groups from 50 yards using Federal XM193 ammunition. (Shooting was conducted at an indoor range, so wind was not a variable in this case.) The two targets are pictured below.The center of the first 3-shot group was located approximately ¼ MOA high and ½ MOA to the right of the center of the target, which was the point-of-aim. However, the center of the second 3-shot group was located approximately 1.5 MOA low and ½ MOA to the left of the center of the target. The centers of just these two, 3-shot groups are almost 2 MOA apart in elevation.The centers of the individual 3-shot groups are indicated below by the blue crosses.I actually fired a total of ten, 3-shot groups in a row with the XM193 ammunition for a total of thirty shots. Thirty occurrences of something that is being studied is what the statistics types like to refer to as a "large sample." Using the RSI Shooting Lab software program I was able to overlay all ten of the 3-shot groups on each other for a 30-shot composite group.This 30-shot composite group gives us a much better picture of the radial dispersion of the rifle/ammunition combination (and in this case particularly the ammunition) than 3-shot groups by themselves do. These 30 shots also give us give us a much better indication of where the center of the points-of-impact are.The ten, 3-shot groups that were fired in a row for the above demonstration measured (from smallest to largest):0.40”0.57”0.64”0.74”0.80”0.85”1.03”1.13”1.19”1.54”The variation between the smallest and largest group above is 74%!In addition to firing the ten, 3-shot groups listed above, I also fired three, 10-shot groups in a row from 50 yards for another thirty rounds of XM193.The groups measured (from smallest to largest):1.24”1.46”1.97”The variation between the smallest and largest of these three groups is only 37%. Using the same total number of rounds in a set, but firing one set using ten, 3-shot groups and the other using three, 10-shot groups shows that the group to group variation of the 10-shot groups isthan that of the 3-shot groups, half as much in this case. This shows again that 10-shot groups are aindicator of the radial dispersion of a rifle/ammunition combination or as Rick Jamison stated, “Ten shots are a more reliable indicator when it comes to predicting what a load is likely to do in the future.”Before I can continue with, I need to define the concept of “mean radius” (also called average group radius.)A Primer on the Mean RadiusThe mean radius is a method of measurement of the radial dispersion of shot-groups that takes into account every shot in the group. It provides a more useful analysis of the consistency of ammunition and firearms than the commonly used method of extreme spread.Mean radius as defined in Hatcher's Notebook “is the average distance of all the shots from the center of the group. It is usually about one third the group diameter (extreme spread)” for 10-shot groups. (The ratio is actually closer to 3.2 times the mean radius = the extreme spread, for 10-shot groups, depending on the sample size and the morphology of the particular groups sampled.)To obtain the mean radius of a shot-group, measure the heights of all shots above the lowest shot in the group. Average these measurements. The result is the height of the center of the group above the lowest shot. Then in the same way, get the horizontal distance of the center from the shot farthest to the left. These two measurements will locate the group center. Now measure the distance of each shot from this center. The average of these measures is the mean radius.Once you get the hang of measuring groups using the mean radius it becomes very simple to do. While being very simple to do, it is also very time consuming. Modern software programs such as RSI Shooting Lab and On Target make determining the mean radius a snap.The picture below is a screen capture from RSI Shooting Lab. The red cross is the center of the group (a little high and right of the aiming point). The long red line shows the two shots forming the extreme spread or group size. The yellow line from the red cross to one of the shots is a radius. Measure all the radii and take the average to obtain the mean radius.Mean Radius DemonstrationLet’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 GroupThe 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 RadiusNow 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 radial dispersion of our rifles and ammunition.Here are some interesting quotes from old issues of American Rifleman on the subject:“Mean radius is the mean distance of bullet impacts from center of the test group. It is used in government ammunition acceptance because it takes account of every shot and comes close to maximizing the test information. While there is no exact relationship between this measure and the simpler and more convenient group diameter, the 10-shot group diameter averages slightly over 3 times the mean radius.”"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.".....Over-laying the targets of at least three, 10-shot groups fired in a row and determining the mean radius of the composite group gives us a statistically powerful tool for evaluating the radial dispersion of a rifle/ammunition combination.....As explained previously, the mean radius method of measuring groups gives us information about each shot in the group, not just the two worst shots of the group as is the case when measuring the extreme spread. The following comparisons will be made using the mean radius of the different groups.Besides doing a 30-shot composite group formed from the ten, 3-shot groups fired in a row with XM193 as shown above, I also compiled a 30-shot composite group formed from the three, 10-shot groups fired in row. Here they are, side by side for comparison.The 30-shot composite group formed from the ten, 3-shot groups has a mean radius of 0.45” (indicated by the inner blue circle.)The 30-shot composite group formed from the three,10-shot groups has a mean radius of 0.42” (also indicated by the inner blue circle.)Finally, I compiled a 60-shot composite group formed from all of the above groups.The mean radius of the 60-shot composite group is 0.44”. This sixty-shot composite group gives us a fairly definitive idea of what we can expect from the rifle/ammunition combination in question when fired from 50 yards. Just as importantly, this demonstrates that the thirty-shot composite groups come quite close to the same results as the 60-shot composite group; differing by only a couple one-hundredths of an inch in mean radius from the 60-shot composite group.....Since the mean radius method of measuring groups gives us a better picture of what is occuring with all the shots in the group, we can use the mean radius measurements to get more accurate comparisons between groups. The mean radii of the ten, 3-shot groups from above are as follows (from smallest to largest):0.17”0.23”0.25”0.30”0.32”0.32”0.46”0.50”0.50”0.69”The 3-shot groups have a variation of 75% from smallest to largest!The mean radii of the three, ten-shot groups from above measure (from smallest to largest):0.34”0.39”0.50”The mean radii of the 10-shot groups only have a group to group variation of 32%, less than half that of the 3-shot groups. Once again this shows the vastly improved consistency of 10-shot groups compared to 3-shot groups.If you recall from the above post, the sixty-shot composite group had a mean radius of .44”. Look at how much closer the mean radii of the individual 10-shot groups come to the mean radius of the sixty-shot composite group than the mean radii of the 3-shot groups do. The mean radius of the 30-shot composite group formed from over-laying the three 10-shot groups on each other is even closer. Consistency and a high degree of predictability are what make the use of 10-shot groups (particularly three,10-shot groups fired in a row) far superior to 3-shot groups in evaluating the precision of our rifles and ammunition!....Here is a final demonstration that further illustrates and reinforces all the concepts previously presented. Using one of my AR-15s that has a free-floated Colt HBAR, I fired five, 10-shot groups from 100 yards using a hand-load topped with 55 grain FMJ bullets. The five 10-shot groups were over-layed on each other to produce a 50-round composite group with a mean radius of 0.70”. The average extreme spread for all five groups was 2.30”.I then proceeded to fire another 50 rounds of the same hand-load in 3-shot groups, for a total of sixteen, 3-shot groups (with an extra 2-shot group at the end.) Several of the groups were sub-MOA.group #06: 0.66”group #12: 0.92”group #13: 0.59”group #15: 0.41”group #16: 0.63”However, several other groups had extreme spreads of well over 2”.group #02: 2.53”group #07: 2.89”group #08: 2.39”group #09: 2.86”group #11: 2.47”Those who espouse the use of 3-shot groups for evaluating the radial dispersion of a rifle/ammunition combination like to point to the smallest 3-shot groups for their accuracy claims and then pretend that all those large 3-shot groups don’t exist.The average extreme spread for all sixteen of the 3-shot groups was 1.72”. Also, the statistical centers of many of the groups, (both large and small) were located at different locations on the target.Pictured below is the first 3-shot group of the 55 grain FMJ load and beneath that the first three, 3-shot groups over-layed on each other.As we continue to sequentially over-lay the 3-shot groups on each, the pieces of the puzzle begin to fill in the gaps giving us a more complete view of the entire picture.Finally, when all of the 50 rounds fired in 3-shot groups have been over-layed on each other, we have a composite group that is remarkably similar in radial dispersion to the 50 round composite group formed from the five 10-shot groups.Here’s the only target from the above shooting session that the Internet Commando would have posted.Remember, all the data for the above examples were obtained during live-fire testing from my bench-rest set-up , not from computed generated models that fail to demonstrate a correlation with actual real-world data.....Institutions and organizations thatenormous amounts of ammunition and weapons are far more interested in facts, than sales hype and propaganda and most of them demand that accuracy/precision testing be conducted using 10-shot groups. This includes the Federal Bureau of Investigation, Crane Naval Surface Warfare Center, the US Army Marksmanship Unit and the US military’s acceptance testing of both 5.56mm ammunition and weapons. On the other hand, businesses that use 3-shot groups for making their accuracy claims are usually trying tosomething.....At the 1992 Barcelona Olympics, USA Shooting Team members Launi Meili and Robert Foth won the gold and silver medals in the three-position rifle events. The Olympians used the new Federal Gold Medal ammunition to aid them in obtaining their victories. This was the first time in more than 30 years that an American won an Olympic medal in one of the small bore shooting events while using American-made ammunition.It’s interesting to note that pertaining to the accuracy/precision development and multifaceted testing of the Federal ammunition that helped the US Olympians win gold and silver medals in Barcelona, Federal’s Director of Product Engineering, Dave Longren, had this to say:* Hunnicutt, Robert. “Ammo Good as Gold.”Nov. 1992: 32-33, 72-73. Print......"We all use math every day. To predict weather, to tell time, to handle money -- math is more than formulas and equations; it's logic, it's rationality. It's using your mind to solve the greatest mysteries we know."Charlie Eppes from NUMB3RS. . . and it helps us evaluate and improve our rifle-craft.Molon.........