Calculating the Figure of Merit

Introduction

I have posted a video showing the process involved in calculating the Figure of Merit HERE.

In a previous blog post I discussed the String Test (here) as a means of judging accuracy in the Nineteenth Century.  In this post we will consider a method of judging precision:  The Figure of Merit.  The British 1870 Musketry Manual calls the Figure of Merit a “measure of the efficiency” of shooting (p. 75), and the 1879 Musketry Manual says it is used to conduct experiments with small arms (p. 314).  Thus, it is not a means of judging shooter accuracy (as is the String Test), but rather the precision of the rifle and cartridges, whereas accuracy is a matter of using the results thus obtained for sight adjustment.  The Figure of Merit system was developed in Britain as rifled muskets changed war forever by making excellent accuracy possible over very long ranges, which led to a need to be able to compare different ammunition loads and different weapons to one another.  The Figure of Merit was later adopted in the U.S. as well; Captain Stanhope Blunt discusses its use for U.S. soldiers in his Firing Regulations from 1889.  

"Part XI--Experiments with Small Arms" of the British 1879 Musketry Manual (p. 314), says that: "The quality of arms, projectiles, and gunpowder are now demonstrated by a figure showing the degree of concentration of a certain number of rounds fired which is called by any of the following terms, 'mean deviation,' 'mean absolute deviation,' or 'mean radial deviation.'"  Interestingly, the Manual says the shots should be taken from a rested position, something not done when judging shooter accuracy alone.  (Note that it does not actually use the term "figure of merit" for this system, however, it is the same process described by the Enfield and Whitworth Report discussed below, and both descriptions use the term "mean radial distance.")  The process calculates the mean radial distance of shots from the center of the shot group and thus measures the consistency of a group rather than its accuracy.  It is more useful than just determining the group size because the Figure of Merit averages the distance of the hits from one another, whereas the group size merely indicates the extreme width of the group.  Thus, two groups might be of the same size, but in one the hits might generally be very close together with one or two flyers, whereas in another the hits might be generally farther apart, and the Figure of Merit takes this into account in a way that looking at the group size alone does not.

Definitions

These terms must be understood in order to follow this discussion:

Point of Aim (PoA):  The point at which you aim your sights; this will not necessarily be where you intend your shots to hit since you may have to “hold off” in order to hit a target for which your sights are not set (e.g., you aim low to hit a target slightly closer than the sight setting, or slightly off to one side to allow for a heavy wind).
Mean Point of Impact (MPI):  The average of the group where your shots actually hit; group center.
Intended Mean Point of Impact (IMPI):  Where you intended your group to center.
Group:  The extreme measure of an entire group of shots.

Procedure

The following procedure for calculating the FoM comes from Report of the Enfield and Whitworth Committee published in April of 1861:
“39. It is necessary here to point out that the following plan has been adopted to determine what is called the figure of merit, or the mean radial distance of shots from centre of group.
40. The horizontal distance of each shot upon the target, from a fixed vertical base, is first found, and a mean horizontal distance obtained.
41. The same process is followed to obtain a mean vertical distance from a fixed horizontal base. The intersection of two lines drawn parallel to, and at distances equal to the horizontal and vertical mean distances from the bases, give what is termed the point of mean impact.
42. The distance of each shot upon the target from this point is noted, and, in the case of those recorded as misses, the distance is taken as equal to one-half the diagonal of the target.
43. The whole of these distances, determined as above, are added together, and divided by the number of shots fired, and the result is termed the figure of merit, or mean radial distance of shots from centre of group.”

(See:  Minshall, David.  "Measuring Accuracy."  Research Press. N.D., http://www.researchpress.co.uk/index.php/firearms/british-military-longarms/small-arms-trials/measuring-accuracy)

For the purposes of calculating the Figure of Merit, consider the following hypothetical target:

Using that hypothetical target as an example, here are the steps necessary in order to calculate the Figure of Merit for this ten-shot string (this procedure is taken from the 1879 Musketry Manual pp. 314-316):
1.  Number each hit on the target so that you can track them carefully.
2.  Measure the distance from the center of each hit to the left edge of the target.  In this case, hit #1 is 2.0 inches from the left edge of the target.  Repeat this for every hit and record each measurement in column X in the table below.
3.  Measure the distance from the center of each hit to the bottom edge of the target.  In this case, hit #1 is 8.0 inches from the bottom edge of the target.  Repeat this for every hit and record each measurement in column Y in the table below.

4.  Misses count as 1/2 of the diagonal of the target.  Since the diagonal of this target is 13.9 inches, a miss would count as 6.2 inches for both X and Y, however, in this hypothetical case, there were no misses.
5.  Average all of the X measurements and record the result.
6.  Average all of the Y measurements and record the result.
7.  These averages give you the Mean Point of Impact.  In the case of the example, the average is 3.55 inches for X and 6.83 inches for Y.  Mark the MPI on your target; in this case the point will be 3.55 inches from the left edge of the target and 6.83 inches from the bottom edge of the target. 
8.  Measure the distance from the center of each hit to the MPI and record each measurement in column Z in the table below.   For example, in this case, the distance from hit #1 to the MPI is 1.95 inches.
9.  Finally, average the Z measurements to calculate the FoM; in this case, it is 1.21.  

Shot	X	Y	Z
1	2.00	8.00	1.95
2	2.25	7.25	1.37
3	3.50	7.00	0.18
4	4.00	8.00	1.26
5	2.75	6.50	0.86
6	4.75	6.50	1.24
7	4.50	7.50	1.17
8	3.00	6.00	0.99
9	3.50	5.75	1.08
10	5.25	5.75	2.01
Avg.	3.55	6.83	1.21

Obviously, this system for determining the Figure of Merit is extremely complicated and cumbersome.  Fortunately, Rob Enfield of the British Muzzleloader YouTube channel has done all the hard work of calculating the Figure of Merit by creating a Microsoft Excel worksheet which does it for you.  The spreadsheet and instructions for its use can be found here.

This spreadsheet is both simple to use and extremely powerful.  It eliminates the need to mark the MPI on your target and to have to measure from the hits to the MPI—the worksheet does all of that for you.  All you have to do is to measure the X’s and Y’s, record them on the sheet, and it does the rest.  In addition, if you enter the other information the spreadsheet calls for it will also calculate the sight adjustments necessary to bring your MPI to your IMPI; this is called “zeroing” your rifle.  The spreadsheet also creates a graph showing all of your hits on a diagram which also includes the Point of Aim, Mean Point of Impact, and Intended Mean Point of Impact.  Here is an example of the graph for the hypothetical target used in this essay using Mr. Enfield’s spreadsheet:

Mr. Enfield posted a video explaining the history of the Figure of Merit and giving detailed instructions for the use of his spreadsheet here.

I have a blog post showing how to use the Figure of Merit to work up a load for a specific piece HERE.

Conclusion

The Figure of Merit is an extremely effective method for analyzing the precision of a rifle and its ammunition because it determines the mean deviation of hits from the group center very exactly.  It has two flaws, however:  First, it is extremely tedious and cumbersome, requiring detailed measurements, records, and calculations.  Second, the Figure of Merit only shows the relationship between the shots and the overall group—it does nothing to show the shooter's accuracy, or how close the MPI is to the IMPI.  When shooting, having all your hits close together is excellent, but ultimately, they should be both close together and close to the point you want them to hit; it does little good to have all your shots group tightly together if you miss the target with all of them.  In fairness, Mr. Enfield’s spreadsheet addresses that to some extent by providing sight adjustment information that can be used to bring your MPI to your IMPI, but that is not, strictly speaking, part of the Figure of Merit.  The String Test measurement mentioned above does both, showing not just the size of your group but also how close that group is to your intended target or IMPI, and it does so using a system which is faster and which requires no detailed records or calculations.  Ultimately, both systems are powerful tools which are superior to merely determining group size, and both are historically accurate.

Range Report: Snider-Enfield Rifle 1/16/21

After months of effort, I finally got to shoot real ammunition (meaning not the Minié balls for fireforming). These cartridges were X-Ring brass and .600-caliber 2-groove bullets with 63.7 grains of Pyrodex P powder (don't give me grief, it's all I can get). I fired 20 rounds at 50 yards from a seated supported position. I used a half sight and aimed at the very bottom edge of the paper.

I confess I am somewhat disappointed. I fired three groups, and got string test measurements of 8.6, 8.5, and 7.7 inches/round respectively. The overall string test measurement (for all 20 rounds) is 8.35 inches/round, which would not qualify me for a Civil War sharpshooter regiment! However, as the attached picture shows, the Mean Point of Impact was 17.5 in. above the point of aim. Now, given that the rifle is zeroed or 100 yards, that's probably not horrible--of course it will hit high at 50 yards. Still, the string test measurements seem high, and are somewhat disappointing; I expected tighter groups since I was firing from a supported position.

On the other hand, I got to shoot a 150+ year-old rifle using ammunition I cast and loaded myself by hand (thanks to Martyn Robinson!!). I love this rifle, and am in awe of it. I don't care if it doesn't shoot minute-of-angle groups, not really, because it's a military rifle and wasn't made for that. I am thrilled to have this chance to shoot a magnificent piece of ordnance. LOL! Now if only I could find more primers and some Swiss powder so I could reload these cases and try to find a better load!!!

I used a wet patch followed by two dry ones to clean the bore after every five rounds. The method I used for loading these rounds can be seen here.

Loading Ammunition for the Mark III Snider-Enfield Rifle

For a video demonstrating how I load these cartridges, go here:
https://rumble.com/v2m3ni6-loading-ammunition-for-the-snider-enfield-rifle.html

Background

The original Snider ammunition went through a surprising nine different marks during the short service term of the Snider rifle.  Most versions had a hollow-base Minié-style bullet with grease grooves.  These bullets also had a cavity in the nose (at first filled with a wooden plug) to shift the mass of the bullet back and to the circumference of the bullet for better stability.  The cartridge cases were formed of brass foil wrapped around a mandrel and then fitted into an iron cup or cups which were held together by the primer assembly.  Ammunition this complex is simply beyond my skill to reproduce, and besides, it turns out to be unnecessary.

Figure 1: The Mark IV Snider Cartridge.

The purpose of hollow-based bullets is to make them slightly undersized so that they can be loaded easily, and therefore quickly, into a muzzle-loading rifle even when the barrel is somewhat fouled.  The explosion of the powder then causes sufficient obturation of the bullet base to fill the rifling completely so that it flies accurately in spite of being undersized.  In the case of a breach-loading rifle, however, a full-sized bullet can be used since it does not have to be rammed down the barrel, and it will still completely fill the rifling by compression rather than by expansion.  This being the case, most modern Snider bullets are solid and of a larger diameter than the originals, as are the ones used here.

 

These original cartridges contained a bullet of nominally .577 caliber (which was actually .573 inches, as the diagram above shows) which weighed about four-hundred-eighty grains (including a wooden or clay base plug as shown in Figure 1) over seventy grains of large-grain rifle powder (plus or minus two grains) and a wad of woolen fiber.

Fireforming

The Snider rifle is only nominally .577 caliber; the barrel is .577 inches, but the chamber of the rifle is actually 0.63 inches while the bullet is .573 inches.  The cartridges cases used for making Snider ammunition today are made from twenty-four-gauge brass shotgun shells and are slightly undersized for the Snider chamber.  In order to resize them without using expensive loading dies they are fired with an undersized bullet in order to fireform them.

 

Fireforming refers to the fact that the act of firing a cartridge will force the case to expand to completely fill the chamber of the weapon, making for a perfect fit for that specific piece.  Resizing dies must be used if you plan to shoot the same batches of ammunition through several different rifles as the process of fireforming will make the brass match only a specific rifle since these antique weapons varied somewhat from example to example.

 

To prepare the cases for fireforming, I prime them using large-caliber pistol primers (see below for details), then fill them with fifty-five grains of black powder along with a sufficient quantity of cream of wheat to fill the case up to the bullet.  I add a twenty-four-gauge nitro card (for shotguns) to separate the bullet from the powder, then I force in a lubricated .575-caliber Burton ball (often mistakenly called a "Minié ball" today); for details of lubrication, see below.  I then take the loaded ammunition to the range and fire it through my rifle.  Afterward the brass is ready to be loaded with the X-Ring Services .600-caliber bullet according to the procedures described in the following.  Note:  These cartridges are horribly inaccurate, so do not expect to hit your target when fireforming—just enjoy the shooting process knowing that things will improve shortly.

Preparing the Cases

Contrary to popular opinion, black powder is not particularly corrosive.  It is, however, extremely hygroscopic, and the water that it absorbs will combine with the salts in the remnants of the powder to corrode brass quickly if not cleaned thoroughly.  These salts are strongly alkaline, so a mild acid will neutralize them.  For this purpose I put the fired cases into a solution of equal parts water and common white vinegar for ten minutes.

 

The next step after neutralizing the powder is to remove the spent primers.  I had a special frame made for various loading jobs, shown in Figure 2 along with the tools used to remove the primers.  The drilled-out circle between the screws on the aluminum plate is for removing primers.  The black rod is the punch used to remove them.

Figure 2: Tools for removing the primers.

The case is inserted into the milled-out spot for it which has a hole drilled in it so that the spent primer can drop out freely.  The punch is inserted into the case so that the point sits into the opening of the primer; it can take a bit of fumbling to ensure that the point of the punch is exactly centered, but it is important to get it exactly right or the point of the punch may be damaged.  A hammer is then used to tap the spent primer out.  Only a few light taps are necessary—if it takes more, then the point of the punch is probably not centered correctly.  Figure 3 shows the punch in place.

Figure 3: Removing the Primer.

Once the primers are removed the cases are ready to be cleaned.  I am fortunate in that I have access to an industrial tumbler used for cleaning and deburring delicate parts.  Simple tumblers can be purchased at any reloading company, but the industrial unit I use can fully clean the cases in approximately fifteen minutes rather than the three-plus hours regular units require.

Figure 4: Industrial tumbler.

The cases come out of the tumbler bright and shiny.  I rinse them with water to remove the cleaning solution from the tumbler, then blow them dry with compressed air, or they can be set aside to dry on their own.  Only the most advanced aficionados use their wives’ dehydrators for drying the cases (inside joke).  Once dry, the cases are ready to load.

Priming

The Magtech brass cases take large pistol primers.  I use the same aluminum plate I used on my loading frame shown above, but any hard metal surface can be used.  The procedure is extremely simple:  A primer is placed on the metal plate as shown in Figure 5.  The case is then set on top of the primer, and a half-inch wooden dowel rod is placed inside the case.  Take care to center the primer exactly in the primer cup of the case before starting.  Next, a hammer is used on the end of the dowel rod to drive the case down onto the primer until the base of the case is flush with the metal surface.  Ensure that the primer is fully and completely seated in place by running your finger over the primer to be certain it does not sit proud of the case at all; if it does, another tap is called for.

Figure 5: The primer and case.

Figure 6: The dowel rod in position to prime the case.

Figure 7: Primed cases.

Bullets

The bullets used for this project were cast using an X-Ring Services #60-530XR bullet mold.  The process of bullet casting is beyond the scope of this essay, but will be covered in full in another blog entry.  One-hundred-percent pure soft lead was used for the bullets (test the lead by gouging it with your thumbnail; if it leaves a mark, the lead is soft enough).  These bullets measure approximately 0.6 inches in diameter and weigh an average of 526.32 grains, plus or minus three grains.  They are solid and have two grease grooves.

Figure 8: X-Ring Services #60-530XR bullet mold.

Figure 9: Fifty .60-caliber X-Ring Snider bullets.

The bullets need to be greased after casting.  I use grease made from three parts beeswax to one part lamb tallow.  I used to pan grease the bullets, but I found that since that only filled the cannelures and didn't leave any grease on the sides of the bullets it made the bullets fit too loosely in the cases, so now I dip the bullets in molten grease and then wipe the bases off on a paper towel before they cool so there's no grease on the base of the bullets to stick to the card wads.

Figure 11:The greased bullets.

Loading the Cartridges

Figure 12 shows all of the components of a Snider cartridge:  The primed case; the one-eighth-inch thick twenty-gauge nitro card wad (from Track of the Wolf); the cream of wheat; the black powder; and the .60-caliber X-Ring bullet.

Figure 12: The components of a Snider cartridge.

I loaded the rounds with 65 grains of Swiss 1.5F black powder; that's a bit lighter than the originals (which had 70 grains of R.F.G), but my rifle is a bit delicate and I don't want to strain it too much.  Brett Gibbons at Papercartridges.com tested Swiss brand powder and found that it gives a muzzle velocity on par with original muzzle velocities, so this is a good analog for the powder used in period.

 

Other Snider shooters include “grease cookies” (a disk of the same grease used on the bullets surrounded by waxed disks of cardboard) between the powder and the bullet, however, Mr. Enfield’s experiments found that these had a negative effect on accuracy, so I elected not to use them.

 

The modified cases are larger in volume than original Snider cartridges and will therefore hold more powder, and in addition I am using slightly less powder than the original cartridges used.  Black-powder cartridges cannot have any air space in them lest the cartridge become more like a miniature pipe bomb than a rifle cartridge, so a filler is required to eliminate excess space.  Soft fillers do not work well with this much space to fill so I used cream of wheat.  Other fillers can be used, but cream of wheat has the advantage of having a granulation very similar to the black powder.  In order to determine how much filler to use I first added a correct measure of powder into a case and measured the distance from the powder to the top of the case.  I then measured the height of the bullet to the top of the second grease groove plus the thickness of the card wad and subtracted that measurement from the empty space in the case, arriving at a measure of three point seven cubic centimeters of cream of wheat to fill the case completely.

 

To load the cartridge I poured the powder into the case using a funnel (see Figure 13), tapping the case on a hard table to fully settle the powder.  I then poured the cream of wheat on top in the same manner.

Figure 13: Pouring the powder.

Next, I forced the card into the case and used a wooden dowel to firmly set it on top of the charge.

Figure 14: The card wad in place.

Once the card was fully seated, I set the bullet into the case and drove it into place with hand pressure.

Figure 15: The bullet in place.

Figure 16: The bullet fully seated in the case.

Finally, I wiped the cartridge with a rag to remove any excess grease left from seating the bullet.  The cartridge was then complete and ready to fire.

 

Note that this method of loading does not involve the use of any dies for resizing, seating, or crimping.  As a result, the bullet is extremely loose in the cartridge case and can usually be rotated in place fairly easily.  This can be disconcerting to people who are used to normal cartridge loading, however, this method is used with complete success by thousands of Snider shooters today without any problems.  In future I plan to use a few drops of melted beeswax to cement the bullet in place just as a precaution and for waterproofing.

Figure 17: A batch of cartridges ready to go.

Packaging the Cartridges

Correct period packaging is important to the historical shooter.  In period, ten cartridges were wrapped in paper printed with a label indicating the contents with a strip of paper woven through them to keep them in place.  Up until about 1870 the cartridges were arranged “head to toe,” meaning the orientation of each round alternated from bullet up to base up.  Later, it was discovered that this orientation cause the rims of the bullets to damage the adjacent soft lead bullets, so the orientation was changed so that all of the cartridges were aligned in the same direction.  Since I am doing an impression from 1868 based on the Battle of Magdala, I elected to use the earlier “head to toe” orientation.

 

To replicate the packaging I made a label based on pictures of extant cartridge bundles which I printed onto letter-sized brown craft paper purchased online because it closely matched the coloring of extant examples.  I then cut a three-and-one-half-inch wide strip off of the printed paper, leaving a five-inch-wide wrapper and a spare strip to use for winding between the rounds.

 

I made a frame three-and-one-eighth inches wide by about five inches to hold the cartridges in place while wrapping them.  I centered the face of the printed wrapper down in the frame and laid the winding strip of paper down against the left side (see Figure 18); I folded up the first inch of the winding strip to make it easier to get it to stay in place as I added the cartridges.

Figure 18: The first round set into the wrapper.

I then added the rounds to the bundle, one at a time, alternating head to toe and winding the strip of paper between them.

Figure 19: The first row of cartridges.

With the first five cartridges in place I laid the winding paper back and added the remaining five cartridges on top in the same way.

Figure 20: The remaining five cartridges.

Next, I folded the sides of the bundle over the cartridges and taped it into place.  Then I folded the sides of the paper up to close it, exactly as one would wrap a Christmas present, and taped the ends closed.

Figure 21: The sides folded and taped.

Finally, I tied up the entire bundle with blue-and-white hemp cord which is a close match to that seen on extant bundles.

Figure 22: A finished cartridge bundle.

Figure 23: Five cartridge bundles complete and ready to issue!

Figure 24: An extant cartridge bundle from 1867 for comparison.  Photo courtesy of Grant Rombough.

Sources

This project would not have been possible without the excellent products from Martyn Robinson at X-Ring Services (xringservices@yahoo.com).  Mr. Robinson offers custom molds for the bullets used here and also the brass cartridges cases.  The cases are made from 24-gague brass shotgun shells manufactured by Magtech by trimming them to the correct length and resizing them.  His products are excellent, and he is an expert on the subject who was willing to answer all of my tiresome questions.

 

The loading procedures discussed above are taken from a series of videos by Rob Enfield of the British Muzzleloaders YouTube channel.  Mr. Enfield’s videos are arguably the best on the Internet and without them I do not think I could have done this nearly so well.

https://www.youtube.com/watch?v=lNgQi4vEDXM

https://www.youtube.com/watch?v=TAEfDVq8cIQ

My sincere thanks to Grant Rombough, a man of great erudition and kindness, for correcting several errors of fact and focus in this essay.