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 x 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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The table below will give you an idea of the relationship between the mean radius and extreme spread for 10-shot groups.



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

“Mean radius is the mean distance of bullet impacts from the 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."



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Excellent post as always.

Very informative!

So, from a reloader's perspective, what is a good benchmark mean radius for match ammo?  Providing a ten shot group at 100 yards.  Is there a military specification?

Just curious...  

I agree that extreme spread alone can be misleading.

I would say at 100 yards, a good benchmark would be a mean radius of 1''.



 

Depends on what you want that rifle to do...

Service/Battle accuracy?
Match Grade?
Hunting?

etc...

How about match grade?

This I think.

Is that a guess or something you have actually researched?  1 inch mean radius (average to center) would make about a two inch group give or take if I am reading this correctly.  That's hardly match grade accuracy.

Incorrect.  The table below will give you an idea of the relationship between the mean radius and extreme spread.

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Molon's calculations come from 3x10rd groups for each load, iirc.


ETA:

Is the 'average to center' (ATC) calculated by on target the mean radius?  I read the definition of mean radius in hatcher's notebook, and it would appear to be the same thing, just making sure I'm understanding it.

Correct.

Excellent questions.  A mean radius of 1.0” for 10-shot groups at 100 yards would be the mathematical equivalent of the accuracy acceptance standard for U.S. Military M193.  As for match-grade ammunition, unlike caliber .30 and caliber 7.62mm ammunition, there has never been a National Match standard for caliber 5.56mm/.223 Remington ammunition.  

In 1965, the caliber 7.62mm Match ammunition was standardized as M118 .  The 1965 lot of 7.62mm M118 National Match ammunition had an acceptance testing mean radius of 1.9” for 10-shot groups fired at 600 yards.  At that time, this was the smallest acceptance mean radius ever achieved for National Match ammunition since records were kept, starting in the year 1919.  Naturally, the ammunition was tested from machine-rested, bolt-actioned test barrels.  The composite target pictured below shows the twenty-seven, 10-shot acceptance groups (that’s 270 rounds!) of the 1965, M118 National Match ammunition fired from the test barrels at 600 yards.  The small circle has a diameter of 6” and the large circle has a diameter of 12”.





From American Rifleman, September 1965





From American Rifleman, August 1962.



Everything else being equal, (which of course, it seldom is) a mean radius of 1.9” at 600 yards would have a mathematical equivalent of 0.32” at 100 yards.  Now, 100 yards is not 600 yards, but then, a semi-automatic AR-15 is not a machine-rested, bolt-actioned test barrel either.  For this reason, I like to use the mean radius of 0.32” for 10-shot groups fired at 100 yards as the benchmark for match-grade ammunition, when fired from a semi-automatic AR-15.  I personally prefer that my match-grade hand-loads have a mean radius “in the twos,” i.e. a mean radius of 0.2x”.  My best hand-loads fired from a semi-automatic AR-15 at 100 yards have had mean radii “in the ones,” i.e. 0.1x”.  The 10-shot group pictured below has a mean radius of  0.13”.

Here is the specification for the SASS MIL-PRF-32316 (AR) :


3 REQUIREMENTS

3.4.1.1 Accuracy/dispersion. After the optic sight is zeroed, the accuracy and
dispersion of the SASS shall meet the following:

3.4.1.1.1 Accuracy. . The distance between the mean point of impact of each shot
group, both unsuppressed and suppressed, shall be not greater than 1.1 inches at 300 feet. The
distance between the mean point of impact of the combined unsuppressed shot groups and the
combined suppressed shot groups shall not be greater than 3.14 inches at 300 feet.

3.4.1.1.2 Dispersion. The average mean radius (AMR) (see 6.11), of each shot group
shall be not greater than to 0.68 inches at 300 feet. All targets shall be fired on using M118LR
ammunition or equivalent, using five (5) round groups.

Hi , I have discover this very interesting topic  meanwhile I was searching for group size measuring methods.
I think that it's very important to have a statistical view of this matter because is different to have a 10 shot group of 5 inchs with 8 holes in 3 inches  tha  a 5 inches group with all 10 shot within 3 and 5 inches.
So i would share this knowledge with an Italian forum  named " Armimilitari" that means " Militaryweapons"  end in order to make this I obviously  ask your permission in advance.
Thankyou

Terrific post.  Thanks.

Done!