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 July 10, 2019, 02:25 PM #26 Unclenick Staff   Join Date: March 4, 2005 Location: Ohio Posts: 16,352 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 cherry-picking 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. __________________ Gunsite Orange Hat Family Member CMP Certified GSM Master Instructor NRA Certified Rifle Instructor NRA Benefactor Member and Golden Eagle
 July 10, 2019, 02:57 PM #27 NoSecondBest Senior Member   Join Date: December 7, 2009 Location: Western New York Posts: 2,342 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
Bart B.
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Quote:
 Originally Posted by NoSecondBest 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.
They think 3 is good enough.

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 fourbore Senior Member   Join Date: August 22, 2015 Location: new england Posts: 1,075 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 Bart B. Senior Member   Join Date: February 15, 2009 Posts: 6,833 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 5-shot 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
Unclenick
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Quote:
 Originally Posted by fourbore A single 10 shot group is ONE data point. N=1 not 10.
Any single group, be it a 2 or a 2000 shot group, is a single data point if the data you are collecting is group diameters or group mean locations of the particular shot count fired. But if the data you are collecting is about individual shot locations, each shot is a datum and n then equals the number of shots. So you need to decide what the data is for. The advantage of staying with individual shot locations is you can always combine them to get larger values of n, and with them, higher certainty about the location of the average for an infinite population of shots. That spot has the highest probability of a shot landing on it, so it is where you want your sights set.

Quote:
 Originally Posted by Bart B. The centers of each of five 5-shot groups are probably not at the same place relative to the point of aim. That can skew the results to smaller numbers.
This is key to understanding the nature of the beast. We already know that no two five-shot groups will be the same size. That's true of any shot count smaller than infinity. It's just that how much they differ increases as the shot count gets smaller, and so does how close the average position of those shots (the mean location) is likely to differ from the infinite shot count mean location. The average location of any shot count that is wandering around will also wander around; just less so than the individual shots because it is an average.

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 CockNBama Senior Member   Join Date: January 6, 2014 Location: Huntsville, AL Posts: 181 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 Bart B. Senior Member   Join Date: February 15, 2009 Posts: 6,833 Picture of the plot of 54 consecutive five-shot groups, ES = ~10 inches: IMG_0220.JPG Last edited by Bart B.; July 12, 2019 at 12:13 PM.
 July 22, 2019, 11:50 AM #34 Cosine26 Junior Member   Join Date: July 7, 2019 Posts: 11 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 Unclenick Staff   Join Date: March 4, 2005 Location: Ohio Posts: 16,352 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 n-1 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 shot-count 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. __________________ Gunsite Orange Hat Family Member CMP Certified GSM Master Instructor NRA Certified Rifle Instructor NRA Benefactor Member and Golden Eagle
 July 23, 2019, 11:05 AM #36 Cosine26 Junior Member   Join Date: July 7, 2019 Posts: 11 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 Bart B. Senior Member   Join Date: February 15, 2009 Posts: 6,833 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
fourbore
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Quote:
 Originally Posted by Cosine26 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
That is an average not the mean. Close enough for govt work and probably about the same value. The exact definition of mean would be the value where 1/2 the shots were inside and other half are outside.
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 Cosine26 Junior Member   Join Date: July 7, 2019 Posts: 11 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,-7-6,4,3 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 Cosine26 Junior Member   Join Date: July 7, 2019 Posts: 11 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,-7-6,4,3 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 stinkeypete Senior Member   Join Date: July 22, 2010 Location: Madison, Wisconsin Posts: 310 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 left-right and the center up-down 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
Bart B.
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Join Date: February 15, 2009
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Quote:
 The practical problem for me is “how does one find the center of a group?”
Group center is the intersection of the average vertical position with the average horizontal position of each shot hole center.

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.

Last edited by Bart B.; July 24, 2019 at 03:58 PM.

 July 24, 2019, 11:24 AM #43 Cosine26 Junior Member   Join Date: July 7, 2019 Posts: 11 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
Bart B.
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Quote:
 Originally Posted by Cosine26 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.
The process and math used a century ago with pencil and paper and a list of math tables is no different than todays computer systems using the same process. Just took longer back then.

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/SPG-55B.

Last edited by Bart B.; July 24, 2019 at 04:38 PM.

 July 24, 2019, 04:49 PM #45 Cosine26 Junior Member   Join Date: July 7, 2019 Posts: 11 For Bart B Try this web site ExteriorBallistics.com
 July 24, 2019, 04:49 PM #46 mehavey Senior Member   Join Date: June 17, 2010 Location: Virginia Posts: 5,310 Hmmmm.... artillery slide rule, firing tables, and charting table.... But I'm showing my age.
 July 24, 2019, 05:20 PM #48 Cosine26 Junior Member   Join Date: July 7, 2019 Posts: 11 Hi mehavy My time in radar goes back to radar and fire control on WWII Pickett cruisers where our main search radar was the SK-2 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
Bart B.
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Join Date: February 15, 2009
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Quote:
 Originally Posted by Cosine26 For Bart B Try this web site ExteriorBallistics.com
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
fourbore
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Quote:
 Originally Posted by Cosine26 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,-7-6,4,3 The median is Seven If you had the individual values you could compute te standard deviation IMHO FWIW
You are correct. Call it a senior moment (longer than a moment, unfortunately) or I am not as sharp as I thought. I will read for a while and let you guys sort this out.

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