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July 10, 2019, 02:25 PM  #26 
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One has to remember statistics of the sort we are using here are meant to estimate how identical rounds will behave in the future. You want to know something about group size because you want to know how reliably you can hit a certain size of target in the future with that same load. They don't lie. Lying with them is done on purpose by people cherrypicking from among statistics to cite one that, in isolation, appears to confirm a bias they want to claim support for. Misunderstanding significance is a way people can fool themselves about what statistics mean, and in shooting it is usually this latter situation that arises.
TL, I think a group's radial SD is a Rayleigh distribution. We should check with member Statshooter, who teaches this stuff for a living. You can also look at a group as a bivariate Gaussian distribution, with one distribution on each of two perpendicular axes, assuming you want to keep negative numbers intact to indicate left or right or up or down relative to the mean. Statshooter always uses 30 round samples and has a good rationale for using this number. I was editing it at one point to put into a sticky. I'm way behind and should finish that and put it up. If you have an infinitely large sample with either distribution, the median absolute distance from the mean (the distance that contains half the population) is at 0.674 standard deviations, so the 7.5" radius shown by Mehavey for M2 should be at 6.74" by my figuring, but they are in the ballpark and you don't expect to resolve all those decimal places without firing closer to a 1000 round sample. In any event, the mean radius has a constant relationship to the SD in an infinitely large sample, so either is a valid basis for comparing groups.
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July 10, 2019, 02:57 PM  #27 
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Working in automotive design and reliability, a sample size of thirty was the absolute minimum size used to indicate any level of confidence. The larger the better. That's one of the reasons I simply can't understand why anyone talks about the size of their three shot groups.....it's pretty meaningless. Ten shots is a much better indicator, and even more if you want significant information.

July 10, 2019, 04:03 PM  #28  
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Most commercial rifle makers use 3 as the limit for their accuracy claims. Most rifles start walking shots away from aim point after 3 due to barrels heating (expanding) up. 

July 10, 2019, 06:01 PM  #29 
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The American Rifleman fires 5 groups of 5 shots per load for 25 shots total with 5 data points. Three shots is not a test.
A single 10 shot group is ONE data point. N=1 not 10. Last edited by fourbore; July 10, 2019 at 06:13 PM. 
July 10, 2019, 07:19 PM  #30 
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I think one 25 shot group would be better, proper barrel fit and stress relief assumed. That is 25 data points in my thinking, all relative to one group center.
The centers of each of five 5shot groups are probably not at the same place relative to the point of aim. That can skew the results to smaller numbers. Last edited by Bart B.; July 10, 2019 at 07:37 PM. 
July 10, 2019, 09:44 PM  #31  
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So, how much will they wander? The statistic that estimates this is called the standard error. It is found by dividing the standard deviation by the square root of the sample size, n. It is the standard deviation of the value of the mean. So if my groups of five have a radial standard deviation of 0.75 moa at my test range, the radial standard deviation of the center of those groups will be 0.75/√5 = 0.34 moa.
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July 11, 2019, 07:37 PM  #32 
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Descriptive statistics, including means, modes, medians, standard deviation and standard error are not answers. They are descriptors. Most especially, they are not predictive.
And the Rayleigh, mentioned twice here, relies on the normal, and is not inherently better for calculating these statistics. I'm not saying you shouldn't take measurements, or pay attention to them, but they don't say all that much about your rifle. They offer a snapshot of you and your rifle, with particular loads on that day. 
July 12, 2019, 09:37 AM  #33 
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Last edited by Bart B.; July 12, 2019 at 12:13 PM. 
July 22, 2019, 11:50 AM  #34 
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Mean Radius
Mean Radius is a method that the US Ordnance Department uses to measure the accuracy of ammunition. It is defined as: Mean Radius: is the average distance of all the shots from the center of the group. It is usually about one third of the group size. To obtain 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. 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 shots is the Mean Radius 
July 23, 2019, 09:31 AM  #35 
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If you've already located the shots in the first two measurements, you don't need the last measurement. Just make the horizontal and vertical measurements for each shot in the same shot order so you have the horizontal and vertical locations of each individual hole as a pair. Subtract the mean horizontal value from each hole's horizontal location and the mean vertical value from each hole's vertical location. When you have those pairs of mean differences for each hole, square them, sum them and find the sum's square root. The result is the distance of that hole from the mean. Take the average of those distances for all your holes to have mean radius. Excel makes all that pretty easy to do.
The basic difference between finding a mean distance and finding the standard deviation of that distance is that standard deviation has you average the squares of the mean differences before taking the square root of them all at once. Also, if you use the sample standard deviation method (mean square divided by n1 before taking the square root) you get a number that makes different sample sizes comparable. With mean radius and population standard deviation (mean square divided by n) the larger shotcount groups average a larger result over the long run, and not the same result. To get radial standard deviation, use the Excel STDEV.S function to get the SD's for the horizontal and vertical mean differences separately. Square and sum the two SD's and take their square root.
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July 23, 2019, 11:05 AM  #36 
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The question was what is the definition of "mean radius" not its validity. At one time the US Army used Figure of Merit in lieu of mean radius.

July 23, 2019, 05:18 PM  #37 
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That picture in post #33 of 270 shots...
Lake City Army ammo plant regularly used 3 or 4 different lots of bullets in one production run of ammo. One or more lots came from bullet making set of dies producing better quality bullets. The other machines, not so good. Sometimes, different jacket metal was better quality in some machines than others. Bullets made in the best set of dies with best quality jacket metal would shoot most accurate. Military teams often pulled the 173 gr. FMJBT match bullets from arsenal 7.62 M118 match ammo then seated commercial match bullets in the cases. Sub MOA accuracy at 600 yards was the norm, sometimes almost half MOA. 
July 23, 2019, 05:37 PM  #38  
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The reason mean is used: image one shot hit the dirt. You measure and now 20 foot has to get averaged into the the calculation where other shots vary from 0 to 8 inches! 

July 23, 2019, 05:56 PM  #39 
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I recognize that this not the mean. I was defining the term "Mean Radius" as used by army ordnance. The question was about 7.62 Match ammo as specified by Lake City.
After reviewing my statics reference: Mean = Average. Defined as the sum of the values of an event divided by then number of events. Median is defined as: the value where the number of values above the median value is equal to the number of values below the median without regard to individual values. In the series below, 19,14,10,76,4,3 The median is Seven If you had the individual values you could compute te standard deviation IMHO FWIW Last edited by Cosine26; July 23, 2019 at 08:07 PM. Reason: Add Data 
July 23, 2019, 07:40 PM  #40 
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Definition of MEAN
Sorry for the repeated post
After reviewing my statics reference: Mean = Average. Defined as the sum of the values of an event divided by then number of events. Median is defined as: the value where the number of values above the median value is equal to the number of values below the median without regard to individual values. In the series below, 19,14,10,76,4,3 The median is Seven If you had the individual values you could compute te standard deviation IMHO FWIW Last edited by Cosine26; July 23, 2019 at 08:10 PM. Reason: Add Data 
July 24, 2019, 10:48 AM  #41 
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The practical problem for me is “how does one find the center of a group?”
In one dimension, it’s easy (I mean, if SDev is considered “easy”). In two dimensions I have to think on this. The “center” of the group could be defined as the point where the sum of the distances from the center of each bullet hole is minimum. Is it as simple as finding the center leftright and the center updown and that’s the center? My concern is that this somehow violates Pythagoras. In a practical sense... isn’t there an app for this? 
July 24, 2019, 11:12 AM  #42  
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Use this method........ Draw a horizontal line through the bottom most shot hole center. Draw a vertical line through the left most shot hole center. Measure and record the distance from the bottom horizontal line straight up to each shot hole. Calculate the average, measure that up from the bottom line then draw a horizontal line through that point. Repeat the above step horizontally from the left vertical line to the right. Group center is the intersection of the above two lines. Interesting stuff: http://ballistipedia.com/index.php?t...adius_.28MR.29 Last edited by Bart B.; July 24, 2019 at 03:58 PM. 

July 24, 2019, 11:24 AM  #43 
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The original question was how is Mean Radius as defined by ordnance report. I gave this info without defending or denying the validity. Perhaps one should return to the time that this was defined. No computers, no calculators, and no Doppler radar, only slide rules and log tables. While it may not be a perfect way to measure accuracy, it is at least a standard. From experience the ordnance department determined that it was a standard that gave a method of comparing accuracy of different lots of ammunition or rifles that was as satisfactory as any other at the time it was developed. Another method was Figure of Merit. Then there was the "string measurement".

July 24, 2019, 02:01 PM  #44  
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I don't think any radar was ever used. Doppler radar is used to calculate direction, speed and range of a moving target; I used to manage people operating and maintaining one type, an AN/SPG55B. Last edited by Bart B.; July 24, 2019 at 04:38 PM. 

July 24, 2019, 04:49 PM  #45 
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For Bart B
Try this web site ExteriorBallistics.com 
July 24, 2019, 04:49 PM  #46 
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Hmmmm.... artillery slide rule, firing tables, and charting table....
But I'm showing my age. 
July 24, 2019, 04:59 PM  #47 
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Hi Bart B
That web sight does not work. Type in Doppler Radar use in ballistics measurement in GOOGLE Try this 2.3.3 Doppler Radar Method Christian Johann Doppler was an Austrian physicist and mathematician who first described the Doppler effect in 1842. He found that when a radio wave, light wave or sound wave is transmitted between objects moving with respect to each other, the frequency of the wave is shifted in proportion to the speed of one object relative to the other. In a Doppler radar system, a transmitting antenna transmits a radar beam toward a moving object. The moving object reflects the beam back to a receiving antenna, which is colocated with the transmitting antenna. Because the object is moving, the reflected beam arriving at the receiving antenna has a frequency that is shifted a small but measurable amount from the frequency of the transmitted beam. This frequency shift is proportional to the speed of the moving object relative to the antennas. In our case, the moving object is the bullet, and the radar antennas are located at the firing position. Doppler radar tracks the bullet as it flies and provides measurements of the radial velocity of the bullet with respect to the antennas; that is, with respect to the firing point. The data from the radar are processed mathematically in a computer using very sophisticated software. At any point in the bullet trajectory, the results of these computations are bullet position coordinates (downrange, crossrange, and vertical directions), bullet velocity components in these directions, and even drag deceleration, all versus time of flight from the firing point. These data are available almost continuously as the bullet flies from the firing point until it impacts the ground. A firing elevation angle of several degrees can be used so that each bullet is tracked continuously as its velocity decreases from the muzzle through the supersonic, transonic and subsonic velocity regions before impact. Knowing the position and velocity of the bullet at any two points along the trajectory makes possible the calculation of a BC value for bullet performance between those two points. Infinity can be used for the BC calculation. The Doppler radar method is far and away the best method of measuring ballistic coefficients, mainly because it provides measurements of bullet performance throughout bullet flight from supersonic velocity levels through subsonic velocity levels. However, Doppler radars are just not readily available. The radar system is very expensive, and a large computer complex is necessary to process the radar data to produce position and velocity data. A crew of several experts is required to operate the instrumentation and process the data. The cost of these capabilities exceeds the affordability limits of all sporting bullet manufacturers, and Doppler radar facilities are available only at some military sites. For the past several years, these authors and other Sierra representatives have been privileged to participate annually for two days in a series of tests conducted at the U.S. Army Yuma Proving Ground near Yuma, Arizona. The Gun Position (shooting site) used for these tests is equipped with a high performance Doppler radar. The facilities are provided by the U.S. Army for tests planned and conducted by the Association of Firearm and Toolmark Examiners (AFTE), which is an association of forensic criminalists from U.S. and international law enforcement crime laboratories. The authors are technical advisory members of AFTE and have suggested tests to be conducted at the Yuma Proving Ground. Measurements of ballistic coefficients versus velocity for a number of bullets of different shapes have been performed over the past three years, and examples will be described in a later subsection Last edited by Cosine26; July 24, 2019 at 05:04 PM. 
July 24, 2019, 05:20 PM  #48 
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Hi mehavy
My time in radar goes back to radar and fire control on WWII Pickett cruisers where our main search radar was the SK2 with a beam width of 20 degrees in azimuth. I was a systems engineer on the TALOS SAM Fire Control system used by the USN. I also worked at the WSMR with the Navy. In those days we worked with the gunners quadrant and as a Navy entity we had to be familiar with the fact that there were two different angular measurement systems. The USN used ~6280 mils to the circle and the US Army artillery used 6400 mils to the circle while the Infantry used 6280 mils to the circle. WSMR was/is a US Army missile test base. Believe me I was one of the first to plank down $400 for a HP 45 calculator and lay aside my slide rule. I aggravated to the HP 34 and the HP 15c programmable calculators. FWIW 
July 24, 2019, 05:21 PM  #49 
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I am familiar with that site's info on doppler radars. While reasonably accurate for bullet speed down range, angular accuracy relative to beam center is not even close to what bullet holes on target show.
I am also aware of what a search for "Doppler Radar use in ballistics measurement" in a web search says. Seen it before. Lake City ammo plant now uses electronic targets to sense bullet position and the maximum error (5 mm, about 2 seconds arc) is not an issue at 600 yards. I don't think any radar at firing point cannot resolve that small of angular error/resolution 600 yards away. Last edited by Bart B.; July 24, 2019 at 09:03 PM. 
July 24, 2019, 06:37 PM  #50  
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