reposted for reference


A PRIMER ON THE MEAN RADIUS

The mean radius is a method of measurement of the 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 (accuracy/precision) than the commonly used method of extreme spread.

The typical method used to measure a group consists of measuring the distance between the centers of the two most outlying shots of a group.  This would be the “extreme spread” of the group.  We are essentially measuring the distance between the two worst shots  of a group.  Take a look at the two targets below.  









Most people would intuitively conclude that the second target shown is the “better” group. Measuring the two groups using the extreme spread method, we find that both groups measure 2.1”.  Once again with the typical method of measuring groups we are measuring the distance between the two worst shots of the group.  This method tells us nothing about the other eight shots in the group.  So how can we quantitatively show that the second group is better than the first?  (Yes,  we could score the groups using “X-ring” count, but this does not give us any differential information about all those shots in the X-ring.)  This is were the mean radius method comes in.  It will give us that extra information we need to better analyze our groups, rifles and ammuntion.  If I just reported the measurements of the two groups above using the extreme spread meathod, without a picture, you would assume that the two groups were very much the same.  Using the mean radius method shows that the second group is much more consistent.  It has a mean radius of 0.43” compared to 0.78” for the first group.

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.

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.

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 make determining the mean radius a snap.

The picture below is a screen snapshot 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 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.









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









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



“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."


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Ummm.... Molon, you just read my mind.

I was just going to look this up..... Thanks.

Been using it for many (40+) years.

It's a howling pain in the ass, but the only statistically way to compare groups.

Make up a spread sheet and plug in the X and Y coordinates of the individual shots, and it gets easier.

ETA: You can draw the X and Y axes anywhere outside the group, just make sure they are perpendicular to each other, and the calculation will yield the same results.

You also don't need to do the third measurement. Individual shot distance from group center can be calculated from the coordinates by using the Pythagorean theorem.

All that shit can better be done with a spread sheet, then all you have to do is measure the individual shot X and Y coordinates from your arbitrary axes.

Awesome post as always OP, keep up the fantastic work!

On another forum I frequent (British Militaria), they often use a number called "Figure of Merit".  I haven't compared the two techniques side by side, but off the top of my head they seem similar.  Are you familiar with the "Figure of Merit", Molon?  Is it different, the same or even just similar?  Maybe just a British way of expressing the same thing?  Anyway, this is great info.

Rob

Not the same thing.  

The "figure of merit" is the mean of the extreme vertical dispersion and the extreme horizontal dispersion.  

In the 10-shot group pictured below, the figure of merit is determined from just three of the 10 shots in the group; thus ignoring all the data provided by the other seven shots.







The 10-shot group pictured above has an extreme vertical dispersion of 3.79".  The extreme horizontal dispersion is 3.22".  The mean of those two figures, and thus the figure of merit, is 3.51".



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Da nada.



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Modern target analyzing software programs now make it a snap.


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(As usual) I agree with Molon on using target analysis software.

If you haven't yet, you owe it to yourself to give it an honest look.

When I was very young and low on the totem pole, I expected to do some of the more tedious lab processes. I made it a point to look for better tools and technology where I suspected it may pay off. Target analysis was one place it was a no-brainer to apply image processing right away. The commercial versions you buy nowadays are just as good or better than what we used back then, and very easy to learn.

If you have learned anything from Molon's tutorial on what the mean radius does for you, then also take the good advice and learn to do this efficiently with software.

It doesn't require much background to learn. If you can point and click, and read and follow instructions at the 3rd grade level, you can get target analysis done with a laptop or desktop using your cell phone camera or dedicated camera very quickly.

Here are one or two resources if you don't know where to start. OnTarget works with or without a scanner, contrary to what the comparison page on targetshootingapp says. You can just take a cell phone snap and scale the photo into ontarget without a scanner.

http://www.ontargetshooting.com

http://www.targetshootingapp.com/wiki/Comparison_of_target_shooting_software

The beauty of setting the X and Y axes on the bottom (or could be top) and leftmost (or could be rightmost) shots is that you eliminate one measurement in each direction.  It's a mathematician's technique for simplifying calculations, and it's beautiful.

And since I don't have the RSI software, like you I'd have a spreadsheet use Pythagorus to get the radii (the scalar of each vector, in fact - that's a built in function in Excel) instead of breaking out more involved manual measurement techniques.  That lets me use a graph paper backing to make measuring each shot's location smoother and more accurate.

Thank you for the excellent information!

I didn't know how this was done before and now I have a new tool for evaluating my performance.

Yup yup yup,
Using mean radius is the ONLY way to know performance.

Wolf Gold .223 55 gr
Nosler Defense .223 64 gr
Federal Gold Medal Match .223 69 gr
Hornady TAP .223 75 gr (not FPD)
CBC Magtech 5.56mm 77 gr

Average to center of 1.051" across all loads.... shot at 100...10 round groups each...2 MOA H1.

Extreme spread wasn't anything to write home about....hence the pointlessness of only counting 10 of 50 shots.
Mean radius is data on every single round fired.

As noted above, modern software makes it a breeze.






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I'm putting RSI Shooting Lab on my wish list...  Now I gotta get a Windows machine running so I can use it.  I've used Macs for so long that I don't have anything running Windows (any version) in the house.  Sounds like yet another project to start working on...

I'm not a Mac guy, but can't you just run a Windows VM on your Mac?

Rob

I'm looking at that.  I've done it with XP, but since XP ain't supported anymore, that's sort of like hiking naked in grizzly country.

I'm thinking my old MacBook Pro will run Win7, and I'll have to check into that.

Molon, the RSI site says you "hold the target up to the screen and click on where the bullet hits are."  Does it let you scan targets too?  That sounds more "accurate" than just clicking on the screen...

Thanks for this lesson

No that I'm aware of.  With RSI you need to use a high quality monitor and meticulously calibrate the software program to your monitor.  You also need to dim the room lighting and orient the monitor perfectly perpendicular to your line of sight to obtain the best possible placement of the "markers" on the bullet holes.

I mostly use On Target these days.  Mostly.


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Great post Molon. I really appreciated the reference to Hatcher's Notebook as I have used that one on a number of occasions over the years, when trying to get a sense of what the group size (extreme spread) was.

B

Gotcha.  Another disc I need to look into...