Accuracy Calculations
How many times have you had someone suggest that you should spend your money to buy a better gun, gun part, or reloading component, or use a top-notch gunsmith, because it will increase the accuracy of your equipment? Or the reverse, where someone suggests spending your money on practice ammo or training aids, because that will improve your scores more than spending money to accurize your gun? I think it's pretty clear to everyone that, in competition, more accurate equipment leads to better scores. But how much better? I mean, would you pay $1000 to increase your score by 8 points in a bullseye 2700 match? How about $1000 to increase your score by 250 points? Those thoughts are nice, but most people talk about accuracy improvement in inches, not points. How many points will a 1" increase in accuracy get you? You need to know that before you can determine whether an expenditure on a 1" increase in accuracy is worth it. But I've never talked to anyone, or seen any post on any forum that really answers that question. If you think about it for a few minutes, you'll realize that's because it is not an easy question to answer. It depends on which target you are using (because they have different-sized rings), and where on the target you are aiming (because the rings on a given target are different widths), and it even depends on what caliber of bullet you are using. That's a lot of variability. So, being a computer programmer, I wrote the program that performs the necessary calculations.Page Contents: Results, How to Use the Results, What is Accuracy?, How does the program work?
Results
The results depend on how you think about aiming. To understand the difference between the 2 different ways to think about aiming (POA = point-of-aim), or how to interpret the numbers in the results pages, see How to Use the Results
How to Use the Results
| The first thing you need to decide is what you mean by point-of-aim (POA). If you say that your POA is "just outside the 8 ring", do mean that, assuming perfectly accurate equipment, the center of the bullet hole will be just outside of the 8 ring (POA=center-of-bullet), or do you mean that it will just miss scoring an 8 (POA=score-of-bullet)? In the former case (POA=center-of-bullet), the score might be a 7 or it might be an 8 depending on the caliber, but in the latter case (POA=score-of-bullet), the score is always a 7. Both ways of thinking about POA have merit. The POA=center-of-bullet idea is useful when thinking about how inaccuracy affects the shots that you will take in the future (because people think about where the gun will be aimed when they pull the trigger). The POA=score-of-bullet idea is useful when thinking about how inaccuracy affects the scores that you got on targets that you shot in the past (because people record their scores, not their center-of-bullet-hole locations). One key thing to remember is that when using POA=center-of-bullet, the scores will depend on the caliber, but when using POA=score-of-bullet, scores do not depend on caliber. Another aspect of POA is "when" is the POA determined? If you want the inaccuracy to include deviations made during the trigger pull, plus the gun, plus the ammo, the POA is where the sights are pointed when the trigger pull begins. If you want the inaccuracy to include just the gun and ammo, the POA is where the sights are pointed when the shot breaks. The program calculates how inaccuracy, measured as a deviation from the POA, affects scores; the program does not distinguish between inaccuracy of the shooter versus inaccuracy of the gun/ammo, and it does not account separately for your wobble area, etc. (P.S. If you want the inaccuracy to include ALL shooter & equipment inaccuracies, then the POA is always the center of the target, and the total inaccuracy is your wobble area + your aiming errors + your trigger pull errors + the inaccuracy of the gun and ammo + environment variances like wind, rotation of the earth, etc. You can easily calculate the points lost to total inaccuracy... it's the total possible points on that target minus your score. The purpose of choosing other POAs is to eliminate some of those variables.) |
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What am I looking at in the Results pages?
Since the volume of result data is rather large, I've broken it into a separate page for each caliber (but don't worry, I've added links to allow you to easily switch back and forth to compare one caliber to another). Each caliber page shows:
- Basic information about the caliber
- Weighted scores for hypothetical 90-shot matches (3 slow, 3 timed, 3 rapid) at different distances and for different POA sets
- Information for each individual target. Each target section has a different background color. For each target, it shows
- Basic information about the target (name, distance, speed (slow/timed/rapid), ring diameters, etc.)
- Weighted scores for a hypothetical 10-shot string into that target for different POA sets
- Weighted scores for individual shots aimed at various distances from center of the target.
When looking a the POA values, "7" means approximately in the middle of the seven ring, "7+" means in the seven ring, but close to the eight ring, "7++" means very close to the eight ring, "7-" means in the seven ring, but close to the six ring, and "7--" means very close to the six ring.
How can you make sense of it all? (Or: Yeah, yeah, yeah, lots of numbers... but what is it good for?)
1) Start by using the POA = score-of-bullet pages. Pick your caliber. Look at the 90-shot match scores for the target set / distance that you're interested in. Look down the first column, and find a POA set that approximates your scores, and then look across that row to see how different levels of equipment inaccuracy would affect your scores on those targets. It can be eye-opening.
2) Want to know whether you should spend your money on practice (buying practice ammo, or buying training tools), or spend your money on gun/ammo accuracy improvements? Use the POA = center-of-bullet pages. Pick your caliber. Look at the 90-shot match scores for the target set that you're interested in. Find a POA set that approximates your current aiming ability. Find the column that approximates your current gun/ammo inaccuracy. The number at that row and that column shows your "current score". Then find the row for the POA set that represents an attainable improvement in your ability (usually up by 2 or 3 rows from your current score), and note the difference from your current score. Then find the inaccuracy column that represents a realistic improvement in the accuracy of your gun/ammo (usually to the left by 1 or 2 columns from your current score), and note the difference from your current score. Which one made a bigger difference? That's the one you should spend your money on.
3) Want to know which centerfire gun will give you the best scores? Use the POA = center-of-bullet pages. Open the caliber page that corresponds to one of your possible centerfire guns. Use either the 90-shot match scores, or the 10-shot target scores. Find the column that approximately matches the inaccuracy of your gun/ammo/flinch at that distance. Then open a second caliber page that corresponds to your other possible centerfire gun. Use the same 90-shot match score table or the same 10-shot target score table. Find the column that approximately matches the inaccuracy of your second gun/ammo/flinch at that distance (noting that it's probably different than your first gun). Then compare the score loss on those two tables for a POA set that approximately matches your aiming ability. That will give you an idea of which gun will give you better scores in centerfire.
What is accuracy, and how do you measure it?
When I started this, I wanted to know how many points better I'd do with a 1.5" gun, versus a 3" gun. That lead to my first problem... what's a 1.5" gun? When a manufacturer guarantees 1.5" accuracy from their .45 wad gun at 50 yards... what does that REALLY mean? Do they mean that if you shot 10 shots from that gun with their specified ammo from a properly-operated Ransom Rest, that you'd always be able to cover/hide all the shots with a cardboard disc that is 1.5" in diameter? Or are they only measuring to the center of the shots (i.e. the holes can be covered with a 1.5 + 0.45 = 1.95" disc)? Or are they saying that, for any given shot, the point of impact can be up to 1.5" away from the point of aim... meaning that the radius of the disc is 1.5", and thus the center-to-center diameter of the group is 3.0", meaning that, to cover all the holes, you'd need a 3.0 + 0.45 = 3.45" disc? How you measure it makes a BIG difference.
As far as I can tell, manufacturers are guaranteeing group size, and the "standard" way to measure group size, is to measure the distance from center-to-center of the two shots farthest apart. This is explained here, and here, and this article gives a more practical, reasoned approach to doing the same measurement. Regardless of how you do it, the standard measure of group size seems to be the diameter of a disc that would cover the centers of all the shots in the group. As such, I would expect a gun that guarantees a 1.5" group at a given distance will consistently produce groups that measure 1.5" or less, center-to-center (which is to say that they can be completely covered by a 1.95" disc, for a 45 caliber gun).
The accuracy (or perhaps more appropriately, the inaccuracy) of a gun/ammo combination can also be thought of as how far the point-of-impact (POI) varies from the point-of-aim (POA). If I hold the gun perfectly still, and aim it perfectly at the center of the target, but the point-of-impact (center of the hole) is 1.5" away from the center of the target, then the gun/ammo/wind/whatever produced an inaccuracy of 1.5". But that's an inaccuracy radius. That will lead to groups that have a diameter of 3", if the gun is held perfectly every shot. Of course, you can't hold the gun perfectly still and aim perfectly at the center every time, so the actual group size from such a gun will be larger than 3" in practice.. But regardless of where the sights are aimed when you pull the trigger, the inaccuracy of the equipment will cause the point-of-impact to deviate from that point-of-aim. The program uses this concept of an inaccuracy radius around the POA to create an inaccuracy circle, and then uses that to calculate a weighted score for each shot.
How does the program work?
Let's say the sights are aimed at the middle of the 7 ring of a B-16 target (the 25yd slow-fire target) when you pull the trigger, and your gun/ammo has an inaccuracy radius of 1.5 inches (this is a 3" gun at 25yds). The 7 ring on a B-16 is 0.75 inches wide, and our point-of-aim (POA) is in the middle, so it's 0.375 inches from the 8-ring, and it's also 0.375 inches from the 6-ring. The 8-ring on a B-16 is 0.61 inches wide, and the 6-ring is 0.95 inches wide. So, if you draw a circle with a radius of 1.5 inches around our POA (call this the inaccuracy circle), part of that circle will be in the 9-ring, part in the 8-ring, part in the 7-ring, part in the 6-ring, and part in the 5-ring. Assuming the inaccuracy is truly random, the bullet is just as likely to be centered at any point inside that circle as any other point inside that circle. Therefore, the likelihood of being in the 8-ring can be determined by calculating what percentage of the area of the inaccuracy circle intersects/overlaps with and the 8-ring. Similarly for the other rings. In this case, there's a 7.99% chance that it will land in the 9-ring, a 19.57% chance that it will land in the 8-ring, 32.05% chance of staying the 7-ring, a 37.40% chance that it will land in the 6-ring, and a 2.99% chance that it will land in the 5-ring. Those percentages allow you to calculate a weighted score of 6.92 (9x0.0799 + 8x0.1957 + 7x0.3205 + 6x0.3740 + 5x0.0299 = 0.7191 + 1.5656 + 2.2435 + 2.244 + 0.1495 = 6.9217, which we'll round to 6.92). So, even though you were aimed squarely at the 7-ring, the inaccuracy of your gun means that (if you performed this same shot many times) you're going to average 6.92 points from it. You just lost 0.08 points to inaccuracy.
How can I use the program results to determine the points I'll gain from an accuracy improvement to my gun/ammo?
Using the example from the preceding paragraph as a starting point... what if you did something to make your gun/ammo more accurate? Let's re-run the calculation on the same shot, but with a smaller inaccuracy radius. Let's go with a 0.75" inaccuracy radius (this is a 1.5" gun at 25yds). Now there's no possibility of it falling into the 5-ring, but there's also no possibility of it jumping up to the 9-ring, either. Now it's got a 17.18% of being in the 8-ring, a 61.09% chance of staying in the 7-ring, and a 21.73% chance of being in the 6-ring. That gives a weighted score 6.95. You lost 0.05 points to inaccuracy.
But it's better than before. Your accuracy improvement just bought you 0.03 points on this shot. Obviously every shot is different, but if you gained 0.03 points on all 270 shots in a 2700 bullseye match, your score would go up by 8.1 points. That could be the difference between finishing 1st and 3rd in your classification, but 8 points is not really a lot for a whole match. Surely, if you're hitting the 7-ring on a regular basis, spending that money on practice ammo is going to improve your score by a lot more than 8 points, right?
However, let's not jump to final conclusions based on a single shot. It turns out that a shot aimed at the 7 ring is one extreme, barely affected by equipment accuracy. Most shots will suffer much worse from the inaccuracy of your equipment. For example, if your point-of-aim is exactly in the middle of that same B-16 target, a gun with a 1.5" inaccuracy radius (3.0" gun) can land them in the X-ring, 10-ring, 9-ring, or even out in the 8-ring, and turns out to have a weighted score of 9.00 points, and you lots 1.00 points to inaccuracy (one reason for the big difference with this shot is because the inaccuracy can never help your score in this scenario, it only hurts). But that same shot with the 0.75" inaccuracy radius (1.5" gun) is always either an X or a 10, and thus has a weighted score of 10.0. So, for that shot, your accuracy improvement gained you 1 whole point, and if you did that every shot for the whole 2700 match, you gained 270 points! Now how much would you pay for that accuracy improvement?
Similarly, shots aimed at the lowest scoring ring suffer badly from equipment inaccuracy because a shot that impacts just outside of that ring doesn't just bring you 1 point down; it's a 4, 5, or 6 point drop, depending on the target. A shot that is aimed exactly 0.75 inches inside the 5-ring on a B-16 target has a weighted score of 4.18 with the gun that has a 1.5" inaccuracy radius (0.59% in the 7-ring, 26.34% in the 6-ring, 51.09% in the 5-ring, and 21.98% of the time you get 0 points because it went outside the 5-ring). But that same shot has a weighted score of 5.12 for the gun that has a 0.75" inaccuracy radius (yes, the inaccuracy of the gun actually improved your score over your POA, because it will never drop lower than the 5-ring, but will sneak up into the 6-ring about 12% of the time). In this case, the accuracy improvement of the gun gained you 0.94 points. Over 270 shots of a match, that's 253.8 points!
The program also accommodates bullet diameter
Also keep in mind that, so far, we haven't considered the diameter of the bullet yet (the calculations in the preceding sections are for a bullet with a diameter of zero). Your point-of-impact is typically considered to be the center of the bullet hole. But your score is based on the edge of the bullet hole closest to the center of the target. So, you get scored for a spot that is half your bullet diameter closer to center than your point-of-impact. So a 22 LR bullet, which has an allowed scoring diameter of 0.2225" to 0.2240" (per the NRA rulebook) gains about 0.1116 inches from this. But a 45 ACP bullet, which has an allowed scoring diameter of 0.450" - 0.454" (per the NRA rulebook), gains about 0.226 inches from this. That's a significant difference, based solely on the diameter of the bullet, and so the program can optionally take this into account, too.
The results show accumulated points lost/gained over 10 shots on a single target, or 90 shots on common "sets" of targets, so you don't have to do the math.
Because the results of accuracy improvement can change so dramatically from one shot to the next, as clearly shown by the examples above, and because they also depend on which target and caliber you're using, it makes for a more meaningful/realistic understanding of how accuracy affects your scores if you look at the points lost to inaccuracy for multiple shots into a given target, or even better, many shots across a set of targets that constitute a whole or partial shooting match. To this end, the program can take the target, the caliber, inaccuracy radius, and a set of POAs as input, and calculate the sum of the weighted score for that set of shots, giving you the weighted score for the whole target. The results pages show the cumulative points lost/gained from 10 shots into a single target, for various hypothetical sets of POAs. These are the 10-shot tables shown for each target on each caliber page.
To simulate a match, the program can take a set of targets that make up that match, plus the caliber, inaccuracy radius, and a set of POAs as input, and calculate the sum of the weighted score for that whole set of targets, assuming the same POAs for each target, and with the inaccuracy radius scaled linearly if the targets in the set are shot at different distances. The results pages show the cumulative points lost/gained from 90 shots into 3 slow fire, 3 timed fire, and 3 rapid fire targets at typical distances. These are the 90-shot tables shown at the top of each caliber page.
The results purposely show various inaccuracies at various targets/distances
Note that a given gun/ammo will have a different inaccuracy radius at different distances. If you're seeing an inaccuracy radius of 1.2 inches at 25 yards, it's logical to assume that you'd have an inaccuracy radius of 2.4 inches at 50 yards. Indeed, that's what the program assumes when you ask for results from a set of targets that are not all for the same distance. But reality isn't always so clean and tidy. For example, if your bullet transitions from supersonic to subsonic during flight after passing 25 yards, you'll find that your 50 yard accuracy will be much worse than merely twice what you see at 25 yards. That's one reason why I kept the 10-shot tables in the results for specific targets, so that you can get a better estimate of what the results would be for your gun/ammo combination at each particular distance.