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What Does 0.5 MOA Mean?

damoncali said:
.... Every discipline has a unique blend of technical skill (ammo, rifle, etc) and shooting skill (sight alignment, trigger control, position, etc). Some are more technical than shooting, others are pretty much all shooting.

Very well said. I might add that winning, although it's nice to win, isn't everything. I love to shoot and I enjoy being in the company of other shooters. I'm not what you'd call "competitive" in terms of the scores I post, but it's OK if I don't 'take home the trophy. I still have lots of fun.
 
sniper 338

when you measure from outside to outside then deduct bullet size , that is not precise and can be very deceiving, measure the whole size you made and you will see a big difference, the whole on paper will be smaller then lets say 308 if that might be what a person is using, center to center with calipers is more exact.
 
Gentlemen, I am a new poster here. Over the past year or so, in my spare time I have developed a somewhat different approach to analysis of precision in grouping based upon the concept of "Circle of Maximum Dispersion," or CMD, i.e., the minimum diameter of a circle which contains ALL shots fired under the same conditions at the same distance. As a corollary to this, the CMD can be used to develop a modification of the "Circular Error, Probable" (CEP) which for this purpose I define as the minimum diameter of a circle which, with 99% confidence, will contain AT LEAST 50% of the shots fired. CEP is a somewhat similar concept as the common "Mean Radius" but is completely independent of group size. It can also be expressed as MOA. I believe that CEP can be a valuable measurement of comparitive grouping capability under constant conditions (ammunition, firearm, and shooter). I have done extensive computer simulation and statistical analysis to establish this concept, and also considerable testing to confirm it, including backtesting against U. S. Navy lot acceptance grouping data for MK262 small arms ammunition. My analysis indicates that at least 200 shot pairs (and preferably more) must be used to establish the CMD and CEP with a very small standard deviation from the mean. These 200 shot pairs are best established by firing at least five groups of 10 shots each, 3 groups of 15 shots each, or two groups of 20 shots, and calculating the mean extreme spread. I prefer use of at least five 10-shot groups as optimum. Groups smaller than 10 shots have been determined to require too many groups to be fired to be practical, as the larger groups are far more data-rich than smaller groups.

If anyone is intersted in the concept, I would be happy to further expand on this discussion. Some may even wish to do additional firing testing using my methodology, which is fairly simple.
 
Dwalt I am interested in your approach, since I am working on a similar aspect which has extensions to predict the effects of chrono and wind std dev. Please PM.
 
If you're interested in the statistics of this sort of thing, someone actually wrote a (short) book on the subject. It's got far more detail than we need to figure out what's shooting well, but if you like statistics, it's got some useful info inside.

I believe it's being sold by a relative of the author in very small numbers just to honor him. Neat little book for the math nerds out there.

http://www.amazon.com/Statistical-measures-accuracy-riflemen-engineers/dp/B0006RA7ZA
 
DWalt said:
Groups smaller than 10 shots have been determined to require too many groups to be fired to be practical, as the larger groups are far more data-rich than smaller groups.

It's my understanding (I'm no statistician) that large groups are sub-optimal when measuring extreme spread because you wind up throwing out so much data - all those shots going into the middle of a big group aren't telling you very much. If you're not going to measure every shot (which is clearly best, and not that big a deal with software these days), I believe the optimal group size is 5-7 shots.
 
damoncali said:
DWalt said:
Groups smaller than 10 shots have been determined to require too many groups to be fired to be practical, as the larger groups are far more data-rich than smaller groups.

It's my understanding (I'm no statistician) that large groups are sub-optimal when measuring extreme spread because you wind up throwing out so much data - all those shots going into the middle of a big group aren't telling you very much. If you're not going to measure every shot (which is clearly best, and not that big a deal with software these days), I believe the optimal group size is 5-7 shots.

Why would you "throw out so much data"? If you shot it, then it's real and tells you something; plus the reality is that the observed variability increases with more observations. This is clear for something like F-class when taking 20-shots vs how well you can do with a 3-shot group. Many of the issues which arise from such discussion boils down to what question(s) are you attempting to answer; and everyone has their own version for this. For me I want to characterize the base capability when I know things are "normal", and I think I should be able to reproduce on a regular basis, and how does that translate into score/group variability when nothing else is going on. Then using this basis, add in the effects of worse velocity variability and the effects of the wind. And, do this in a simple manner using ordinary software such as Excel. Of course this can be approached from much a more complex method, but then the weak link always comes from the less predictable elements such as wind, mirage, etc.
 
On the contrary, large groups tell you much more about grouping capability than small groups. The ES variability for a 3-shot group has a standard deviation around a mean ES of about 36%. For a 10 shot group, the SD is about 11%. For a 20 shot group, the SD is about 4%. So the ES for larger groups is far more consistent, group to group, than for small groups. It all has to do with shot pairs in the group. A three-shot group has only three shot pairs. A 5 shot group has 10 shot pairs. A 10 shot group has 45 shot pairs (equivalent to that of 15 3-shot groups). For 20 shot groups, it's 190 shot pairs. The formula is: Shot pairs = (N/2) X (N-1) where N is the shots per group. So, large groups are vastly more data-rich than small groups. As the group size (number of shots) increases, its ES approaches the "Circle of Maximum Dispersion" (CMD). For example, the CMD of an average of a very large number of 3-shot group ESs has a multiplier of approximately 2 (see below), i.e., if your mean ES for a large number of 3-shot groups is 1" at some distance, the CMD will be 2" at the same distance. The same scenario for a mean ES for a number of 10 shot groups has a multiplier of about 1.3, i.e., if the average ES from a series of 10-shot groups is 1", the CMD is about 1.3" For a 100 shot group, the multiplier is about 1.05. The key point to remember is that the CMD is a constant for a given gun-ammo-shooter system, regardless of group size (shots/group) used. But CMD can be far more efficiently, easily, and precisely estimated from firing fewer large groups to get more shot pairs, and more consistency. And the CEP will always be approximately 60% of the CMD.
As previously stated, to get a good mean ES for any group size, the total number of shot pairs fired is preferably at least 200, and the more, the better. For example, firing five 10-shot groups is 5 groups X 45 shot pairs per group = 225 shot pairs. Two 20-shot groups represents 2 x 190 = 380 shot pairs, even better. In any event, the standard deviation of the CMD (and the CEP) will be less than 5% of the value if 200 total shot pairs or more are used. You can see how impractical it is to fire 70 three-shot groups to get the same reliability as firing five 10-shot groups, and uses far less ammunition.

CMD ES Multipliers: Average ES x Multiplier = CMD

3-shot groups - 1.93 (to the nearest 0.01)
5 shot groups - 1.59
10 shot groups - 1.30
15 shot groups - 1.17
20 shot groups - 1.15
30 shot groups - 1.14
50 shot groups - 1.10
100 shot groups - 1.05
---------------------------------------------------
So, let's go back to the question of the rifle that shoots 0.5 MOA "all day". If you interpret that standard as meaning it will put EVERY shot fired into 0.5 MOA, that means that the CMD must be 0.5 MOA. Therefore, assuming you shoot five 10-shot groups and average the ESs of all five groups, the five-group mean ES would have to be 0.5/1.30 = 0.385 MOA before you could brag about having the elusive rifle which would shoot 0.5 MOA all day. Anything greater would not be a 0.5 MOA rifle.
 
You throw out data because you're only measuring the two most extreme shots. All you know about the rest is that they were better - but not how much better. What you really want is to measure every shot.

In other words, two five-shot groups is better than one ten-shot group.
 
Aside from the fact that neither two 5-shot groups nor one 10-shot group can establish anything of statistical significance about grouping performance, a single 10-shot group has over twice the shot pairs of two 5-shot groups (45 vs. 20), which is a 2.5X increase in available data. Therefore, an ES from a single 10-shot group is a much more reliable indicator of grouping performance than the average ES of two 5-shot groups. An ES of one 20-shot group is very nearly there in terms of being an excellent indicator of grouping capability. Those shots "in the middle" are not in any way wasted. They are simply the results of shot distribution, which is necessary to establish the ES. I have seen that reference to "wasted shots" many times in many places, yet, if you think about it, how would you get an ES without them? I know the idea behind measuring all the shots to calculate the Mean Radius, yet the Mean Radius tells you nothing more than the ES will, as the two are very closely interrelated.
 
I am in great danger of wading in over my head here. But aren't you assuming a specific distribution when you say that the interior shots aren't wasted? Is that a good assumption? Honest questions - I'm not trying to be argumentative.

Also, I'm curious to hear what you think of this:

http://www.geoffrey-kolbe.com/articles/rimfire_accuracy/group_statistics.htm
 
When I analyzed F class targets, the results were clearly normally distributed. I have read other distributions postulated, which may in fact apply to something like 100 yd benchrest where the shots are on top of each other. So measuring the distance for every shot allows one to use the standard deviation and the well known normal distribution to make inferences along several fronts. For example target scoring whereby points progressively decline as the poi is further from the bull is directly analagous to describing the distribution of results. Predicting scores-distributions is straightforward. Predicting groups is not a single answer, since these results also follow a distribution.
 
The point is not so to win matches. It's to save money, time, and components by efficiently developing a good load.
 
Well from what I have read you're shooting 200 rounds, most might shoot 50 or 60 have a load you can win with, it's not rocket science, run a ladder test 3 shots take the best shoot 5 shot group and you got a load, there are big big winners in here that do just that one just recently had a big win 1000 yard f-class by simply doing that Erik Cortina, that is just one of many big winners, my simple way got me 3 wins and I showed up 3 times 2 first place and one second place, but if yoo're enjoying what you're doing then that's a good thing but it's not necessary to compete or hunt, you can read article after article of the world class shooters in here and you will see what they do.
 
It's certainly not rocket science, and borders on the academic. I don't spend anywhere near 200 shots in load development. Maybe 100 at most, depending on how early tests go. But I like to spend those 100 shots in a manner that gives as much confidence as possible that what I'm seeing is real.

Ladder tests, OCW, and similar stuff as reported on boards across the internet just don't do that very well. It's a waste of ammo if you ask me. You have to shoot real groups - the number and size of which are apparently debatable by people much better at this than I am.

That said, the truth of the matter is that my last F class barrel shot just about everything I put through it well. Best load was sub .5 MO the worst was about .65 MOA. Had I accidentally chosen the wrong load because of incomplete statistics, it would not have caused a significant disadvantage. I think a lot of "top shooters" do exactly that and still manage to win. But why not do it right if you can figure out how? It costs nothing other than a little thought.
 
In response to damoncali's posting, the referenced article has the theory approximately correct. My approach is somewhat different, in that what I did was to develop a computer simulation by generating a series of random "shots" using a standard distribution, all "shots" (actually X,Y coordinates) were restricted inside a unit circle, i.e., a circle having a radius of one unit. This involved the generation of thousands of "shots". Then the distance of every shot from every other shot was calculated, creating an enormous array of hundreds of thousands of shot pairs, ranging in length from zero to 2 units. Then groups of various sizes are created. For example, a five-shot group consists of 10 shot pairs. So 10 random shot pairs are captured, and the largest pair length becomes the ES. THis process can be repeated to infinity if desired (I didn't go that far). The same thing can be used for 10-shot, 15-shot, 20-shot, or however many shots are desired for a particular group. Remember that the larger the number of shts per group, the more shot pairs are involved. For example, a 20 shot group contains 190 shot pairs. Then you can create multiples of the shot groups, for example, you can create five 10-shot groups, and average the ESs of each of the five groups. Again, this can be replicated as many times as you wish. This process is called a "Monte Carlo" simulation. Any number of statistical analysis operations can then be performed on the results. Remember that every group cannot exceed an ES of 2 units (due to the unit circle restriction), so that becomes the "Circle of Maximum Dispersion" (CMD). This allows the calculation of the CMD multipliers, and many other parameters. I won't go any further into the exact methodology, but once you get into it, it's not that complicated if you know how to work with Microsoft Excel. The important result for the present is just how many groups of each size are necessary to get a very consistent estimate of the average ES for every size group and how many groups are needed to get a very low standard deviation (SD). In my case, I assumed that a SD of no more than 5% of the mean ES would be satisfactory. The results:

Shots per group Number of groups to be averaged for 5% SD

2 about 250
3 about 75
4 about 40
5 20
10 5
15 3
20 2

As you can see, the most efficient group sizes (in terms of how many shots are required to get statistically reliable results) are five 10-shot groups (50 shots), three 15-shot groups (45 shots), or two 20-shot groups (40 shots). Actually, a single 20 shot group is very close to the 5% SD boundary. My preference is the use of five 10-shot groups, as smaller group sizes are very inefficient. I have personally tested the results on the range extensively using a Winchester Model 52 match rifle at 50 yards with match-grade ammunition, and was provided a large amount of U. S. Navy lot acceptance data for the 5.56mm MK262 (Black Hills) round fired at 300 yards from test barrels to analyze. The Navy has adopted the five groups of 10 shots each as their grouping measurement standard, but has taken it further by firing five 10-shot groups from each of two test barrels. If grouping results for either barrel is below their acceptance standard (4" ES), the lot is rejected. So far, no lots have been rejected. (FYI, the ES of the 10-shot groups run about 2.5" or less, rarely above 3", i.e., true "1 MOA, all day"). My analysis of the Navy acceptance data verifies the 5% standard deviation, at least in the average of all lots I examined. Several lots were only slightly higher.

If one wishes to perform a meaningful comparison of precision during load workup, I submit there is no better way to do it than firing at least five 10-shot groups and determining the average ES. It will also provide considerably more information such as good estimates of CMD and CEP, which is, I believe, superior to attempting to measure mean radius. An example:

Five 10-shot groups of a given load are fired at 100 yards. The ESs of each group are averaged, the average ES being 0.75". For 10-shot groups, the CMD multiplier is 1.30, and the CMD is 0.98" (0.75 X 1.30). This means that your rifle and load combination under constant firing conditions is capable of placing EVERY shot into an approximate 0.98 circle at 100 yards (or if you prefer, 0.93 MOA) under identical conditions. If you put much faith in "Mean Radius," you can then use Circular Error Probable (CEP), which is analogous. CEP is the diameter of a circle within which there is a 99% confidence that at least 50% of the shots will strike. My analysis indicates that CEP is 60.4% of CMD, but 60% is OK. So 0.6 X 0.98 = 0.59" (or 0.56 MOA). As statistics does not deal with precise numbers, but probabilities and deviations, it's safer to say that your rifle and load is capable of delivering a CMD of 0.93 and a CEP of 0.56 MOA within +/- 10%. That's pretty good for comparing load performance. Note that all the extraneous factors such as weather conditions, temperature, and wind are not considered in this analysis. If you want more confidence, simply fire more than 5 groups to establish the mean ES.
 
DWalt, Chris Long (Optimum Barrel Time author) did a similar simulation a while back to show similar results; the study is on his home page. The interesting aspect is that to answer many what-if's you do not need to run simulations when utilizing a Gaussian, normal distribution, since its behavior is well known and documented. When comparing the results of Chris's simulations I found his factors are directly correlatable to those routinely used for daily statistical work such as SPC.
 

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