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March 10, 2010, 11:40 AM  #1 
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What is the signifigance of Standard Deviation??
Hi everyone,
Got my hands on a chronograph yesterday to test some new loads. One of the features that it gives is Standard Deviation. Most of my 10 shot strings for most of my loads fell between 12.8  3.1 standard deviation. Im not sure what this means. Can anyone help???? Thanks!!! George ps: the chrono i used was a friends Competition electronics with the infrared setup. worked flawlessly, even with rapid fire strings!!!! 
March 10, 2010, 11:51 AM  #2 
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March 10, 2010, 12:01 PM  #3 
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The way I think of Standard Deviation is essentially the average amount that the average shot deviates from the average.
Confused? In complex terms: http://en.wikipedia.org/wiki/Standard_deviation In simple terms: You might have muzzle velocities of: 2800 2805 2810 2790 2800 for an average (mean) of 2801 and a standard deviation of 7.4162. or, you might have muzzle velocities of: 800, 2805, 2810, 2790, 4800 for an average (mean) of 2801 but a standard deviation of 1414.23 In other words, more or less, the "average" shot of the first string is "off" the average (mean) by an "average" of 7.41fps while in the second string, the "average" shot is "off" the average (mean) by 1414.23fps. This is not the truest mathematical explanation, but it is essentially what the SD means to "average" people.... How much can I expect any given shot to deviate from the average.
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March 10, 2010, 12:06 PM  #4  
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Quote:
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March 10, 2010, 12:25 PM  #5 
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We studied standard deviations in a university statistics course, and never used it again  if you're not a statistician, don't worry about it. Here's a site with a great explanation (I like the animated graphic) link.

March 10, 2010, 12:38 PM  #6 
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In a reloading situation you're trying to minimize your SD. A big SD can show that you're not reloading consistently or you're having issues with pressure jumps, for example due to inconsistent ignition using standard primer where magnum is indicated.
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March 10, 2010, 12:51 PM  #7  
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The difference in velocity from shot to shot is your "deviation". Usually measured with a string of 5 shots (sometimes a few more). SD gives you a measurement of how much each of your shots varies. A SD of zero would indicate you did everything exactly the same between cartridges (unlikely). If your SD is a low number, you'll have consistency from shot to shot. A "good" SD vs. "bad" SD varies from cartridge to cartridge and between rifle and handgun rounds. It can even vary between brands of brass (some have smaller volumes than others). Clear as mud?
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March 10, 2010, 01:16 PM  #8 
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Under a standard bell shaped curve, where most of your shots occur at the apex of the curve, SD is a measure of how far off from the apex of the curve you can expect your shot to be. For example, 1 standard deviation brackets a small area on both sides of the apex, while 2 standard deviations brackets a larger area etc..
The lower the number, the better off you are, as far as consistency goes. 
March 10, 2010, 03:46 PM  #9 
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Wiki has a good explanation.
http://en.wikipedia.org/wiki/Standard_deviation In theory, if what you are doing follows a bell curve (and not all things do!), then 68% or the time the velocity should be one standard deviation from the average. In a practical sense, comparing SD of loads is not as important as on target accuracy. Also, I consider the extreme spreads to be important in sorting out powders. For handguns, when you get extreme spreads in terms of high 100’s, 200’s, or god forbid more, the powder is inappropriate for the application. 
March 10, 2010, 03:52 PM  #10 
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On the chronograph, deviation is how far a particular shot's velocity is from the average velocity of all the shots in the string. Thus, a particular shot whose velocity is 10 fps lower than average and another particular shot whose velocity is 10 fps faster than average both have the exact same deviation: 10 fps. Note that deviation has no sign (no plus or minus). Standard deviation is just the average of all the individual deviations of shots in the string.
The formula commonly given for standard deviation is more complicated looking than the above. That is because, in order to find the deviation you first have to subtract the average from each individual shot velocity, which gives you a minus sign half the time. You then have to remove the signs. If you don't remove the signs, the result of averaging them will just be zero rather than the average deviation. To figure the standard deviation algebraically, the deviations are squared to kill the sign, then the square root is taken to return the absolute value or magnitude which has no sign. The common standard deviation formula given to accomplish that is the result of algebraic manipulation to reduce the calculation to a form that takes the fewest steps to solve in a calculator or in Excel. But most people can't just look at that formula and intuitively grasp what it is getting at? That causes confusion and makes the concept look complicated. If your velocities are truly randomly different from the average value, and if you fire enough of them, you will find the velocities can be plotted on a histogram that forms a Gaussian probability distribution commonly familiar as the bell curve. Because that function goes to infinity at the ends, it turns out that over an infinite number of shots you would find 68.2% had velocities equal to or closer to average than the standard deviation in feet per second. The rest would have bigger deviations than the standard deviation. Wait a minute, you say! Wouldn't half the shots be above average and half below? No. This is average deviation, not average value. The bell curve is center weighted heavy. Each small deviation affects the average deviation less than a large one does, and it turns out the less numerous large deviations fly enough further out that it only takes 31.8% of them to pull the standard deviation value to where it is. So what use is all this? Often, not much. The idea of calculating standard deviation is based on the assumption that velocity deviation is random, and not caused by, say, using two different primers mixed up in the loads you are testing, or two different case brands, or some other nonrandom source of deviation. The idea is that it tells you how broad the bell curve is around the center (the mean, or average value of your velocities). That tells you how consistent your load is and from it you can calculate the probability that a shot in any given group will be enough different to cause a significant error in point of impact on your target? The problem with the above is usually sample size. You have to fire enough shots so the deviations you get are relatively symmetrical around the average. That is, if you put the signs back in, the negative (lower velocity) deviations by themselves would average a value that, when you removed the sign, was not significantly different from the average for the positive (faster) velocity values by themselves. Without that symmetry, the center of the bell curve for an infinite number of such shots isn't likely to be reflected very accurately by your actual shot string's average. That last point is where most shooting measurements run afoul of statistics. A rule of thumb used by statisticians is that it takes a sample size of about 30 to have reasonable confidence that the bell curve is symmetrical and that the average is close to the average an infinitely large sample would give you. Without that, probability predictions for your ammunition performance based on the bell curve are not likely to be usefully accurate. How much velocity variation matters to you at all is going to vary with your purposes and accuracy requirements at the longest range you expect to shoot. But there are other reasons to record it. For one, it affects barrel time, as I mentioned, so, in some instances it has a disproportionate effect on group size not because bullet drop due to different velocity is causing it, but because you get into a bad place in muzzle deflection under recoil. It is likely you will discover a velocity window for a given combination that is best, and standard deviation is a good way to define the window. That is because defining the bell curve projects deviations not yet actually fired. You can also use this to compare component lot changes. Does a new batch of primers or bullets or cases or powder replicate the previous lot well enough to produce the same accuracy? Nick
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March 10, 2010, 08:57 PM  #11 
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Good info, Nick.
What do benchrest shooters consider an acceptable SD? 
March 11, 2010, 01:02 AM  #12 
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Standard deviation is simply a way to quantify spread of a series of data points, like velocities.
It's better than the extreme spread because one bad round can make the extreme spread pretty...extreme. But standard deviation accounts for that. In this context, it's simply a way to define how consistent in velocity your rounds are. (In statistics, using a distribution that is normally distributeda normal curveabout 68% of all data points will be within 1 standard deviation of the mean. About 95 percent of all data points will be within 2 standard deviations of the mean. But this is for a population of data points whose distribution approximates a normal curveor for the standard error of the mean, but you don't want to go there). I focus on SD as a basis for consistency. It's more stable as a measure, takes into account the extreme shots if any but not overly so. I've read that under one standard deviation for rifle rounds indicates pretty decent consistency (such as a SD of 28 when your average velocity is 3100). I've generally been able to achieve that w/ .223. I can't achieve that with handgun brass, in part because I'm using manyfired brass of varying headstamps. I consider 2 standard deviations to be reasonable given the handgun ammo I'm shooting. Finally, the number of data points matters. I generally don't pay any attention to SD unless I have at least 10 shots in a string. It will give me a good idea, but not perfect idea, of how that load tends to be in terms of consistency. I'm testing some new bullets tomorrow. I'll try to shoot them through my chrono (assuming the range isn't too busy). But I only have 5 or so of each load, which I'm primarily testing to ensure I am at reasonable pressures/velocities. Once I'm comfortable that I won't be out of bounds with these, then I'll do a more comprehensive set of loads for accuracy testing. I won't care about the SD much except that I want those shots to group around the mean in a reasonable mannersay within a couple SDs max. But if I shoot 3 or 5 rounds, that's not a sample size that'll be reliable for me. Last edited by mongoose33; March 11, 2010 at 06:39 AM. 
March 11, 2010, 09:38 AM  #13 
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I think I'll Standard Deviate some 270's today.

March 11, 2010, 12:55 PM  #14 
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TexS,
It depends on the range, but they like it as small as possible. The guys going for 600 yards and out try to stay within an SD of 10 fps if they can manage it? Nick
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April 3, 2010, 03:30 PM  #15 
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My current reloading project is to find the "best" 165,000 PF load for my SW625 45 ACP moon clip revolver using 230g LRN bullets and the gunpowders that I have on hand.
After reading a fair number of posts about chronometer results and statistics I've come up with a plan. Hopefully more experienced reloaders than me will comment if I'm zigging where I should be zagging with this. First I'm looking at the velocity extreme spread (ES) and mean velocity for a small batch of a test load. For example one test batch using 700X looks promising: mean velocity 741 fps with ES of 25. On the other hand two different Green Dot loads got ES values respectively of 70 and 84 which seems high. Maybe this is telling me that Green Dot isn't particularly good for this application. Eventually I hope to find some loads that warrant by their small ES numbers testing a 30 shot batch so as to develop a good SD statistic. That 30 count comes from my reading. In the end a good load might look something like this: mean 780 with SD 20. That mean minus 3*SDs gives 720 which makes 99% of the time my PF goal. That mean plus 3*SDs gives 840 which is under my max of 850, a somewhat arbitrary number but I think reasonable. I'll want the SD to be as small as possible and this suggests it should be no more than about 20. The final test presumably is measuring how precisely the load shoots. What are the chances that a consistently shooting load by velocity will be problematic for precision at the target? I'll eventually find out! 
April 3, 2010, 04:18 PM  #16 
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Welcome to the forum.
ES doesn't have much affect on accuracy at close range. Bullet weight variance will have more effect. The reason is that there is a kind of selfcompensating effect at close range. A faster bullet is fired with greater recoil which causes faster muzzle rise, but it gets out of the muzzle in a shorter time frame, so it ends up clearing the muzzle with about the same muzzle elevation as it would have had with a lower charge. Try loading some rounds with 10% difference in charge weight and shoot some groups off bags with them at pistol target ranges to see how much difference it makes in your gun? Up to 25 yards, in particular, I think you'll be surprised how little difference there is. But try keeping the load constant and changing the bullet weight 10% and you will have a very different story. To keep SD down for qualifying purposes over the match range officer's chronographs, several strategies may help: Pick the right powder. Some are more position insensitive than others. You want a position insensitive powder. In .45 ACP, powders that fill the case well often are too slow to generate enough pressure to burn consistently, so don't expect case fill alone to be the determining factor. Your 700X may be doing this just fine for you. So might Bullseye. So might Hodgdon Universal. Spherical propellants like 231 (a.k.a., HP38) or Accurate #2 will be easier to meter consistently and may do all you need as to position insensitivity? Accurate number 2, in particular, boasts of position insensitivity in its product description.
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April 3, 2010, 06:27 PM  #17 
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SD is a useful statistic.
Things that are + 1 SD are statistical outliers (or rare events), especially if they are +2 SD. If you were to shoot thousand and thousands of rounds (and I hope you do) and measured all velocities (I hope you don't) and charted them you would have a distribution resembling a normal bell curve around the mean. Around 68% of all cases exist + 1 SD. About 95% of all cases would exist + 2 SD. Small SDs indicate that you are being very precise with your reloading method. Big SDs may indicate a problem. If for instance I were to find that my SDs were larger with WIN brass and they were smaller with RP brass    well I would prefer to buy RP brass... Same with primers.... Same with powder... Bullets... It is about making a predictable, repeatable, process. So yes I think SD is very important.\ ps: and now that I'm done typing this I read mongoose's post   which is spot on. 
April 3, 2010, 08:30 PM  #18 
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Actually, 1 SD is only about 80% of average deviation, owing to the root mean square finding part of the calculation (squaring to be rid of signs, finding the average of those squares, then taking the square root of that average).
Where the area under the Gaussian bell curve is 100% of the the population of an infinite number of samples, the portion of the population under the curve inside the SD limits to two decimal places is: Within 1 SD of the mean, the area under the curve will be 68.27% of measured velocities Within 2 SD of the mean, the area under the curve will be 95.45% of measured velocities Within 3 SD of the mean, the area under the curve will be 99.73% of measured velocities Within 4 SD of the mean, the area under the curve will be 99.99% of measured velocities Because of the few extreme outliers (the bell curve goes from minus infinity to plus infinity) the average deviation (mean deviation (MD)) either side of the mean is just over 1.25 standard deviations. The portion of the population under the bell curve lying inside the MD boundaries is: Within 1 MD of the mean, the area under the curve will be 78.99% of measured velocities Half the population: Within 0.6745 SD of the mean, the area under the curve will be 50.00% of measured velocities All that assumes the shot distribution fits the Gaussian bell curve of normal distribution, and there aren't some nonrandom boundary conditions like mixing two different loads into the sampling.
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April 12, 2010, 10:26 AM  #19 
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Thanks for the suggestions. I have purchased a Redding profile crimp die. their reputation is to put a really firm hold on the bullet. I read in 45 ACP bullseye discussions that an 0.469 crimp is good so I'll try that first. I also bought a 45 ACP case length gauge. For subsequent chrono work I'll sort the brass and only use the cases that are on spec. Lastly I have some Federal 150 primers. My S&W 625 "Miculek" revolver will double action lightstrike on some primers; I've not had that problem with "150" primers.
Regarding the statistics I've been trying to so some homework but mostly I've just beat my head against a wall of complicated theory. I want to understand what sample size is needed for this application to generate a good SD. I've read that 30 observations will work. There must be underlying assumptions leading to that value "30." Perhaps the number of observations needed is a function of the variability (although it is the variability that I am trying to measure). Maybe this is an iterative process. 
April 12, 2010, 11:54 AM  #20  
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In your case, it's irrelevant. If you're really interested in laying this out statistically, consider using a confidence interval for the mean velocity. I teach statistics. I evaluate my handloads using a chrono, and I consider 10 rounds to be enough of a sample to give me a good idea what a particular load is doing. To me, a chrono tells me where a particular load tends to center in terms of velocity. If the standard deviation is very large, it tells me the mean velocity is not particularly stable, i.e., my estimate from the average is based on some pretty wildlydisparate observations. Which statistic, or which method you wish to use to evaluate the consistency of your ammo depends on what you're trying to accomplish. When I use my chrono for working up loads, my initial rounds through it (usually no more than 5 for a starting load) are simply to establish that I'm in a velocity (pressure!) range that I'm comfortable with. Then, as I add more combinations (again, in the lower end of the loads, 5 rounds is usually it), I'm checking to see how velocity/pressure is doing. If you're careful in how much powder is going into each case, and careful as to maintaining OAL, I don't believe you need more than 5. You're simply establishing safety parameters. But as I get closer to max or what I think will be my production load, I'll load up 10 (maybe 15), and shoot those. I think 10 is just fine for establishing how the round performs, more or less. If i have a weird flyer (something that expands the ES a lot), I'll consider it bad brass or something else. But mostly that doesn't occur. Think of it this way: If I work up a load with 5.7 grains of W231 powder at 1.225 overall length and produce velocities of 857 846 832 864 859 849 852 848 791, it's easy to see that the last velocity is not typical. If I want to describe what a 5.7 grain load of W231 does, is it reasonable for me to exclude the 791 and say that I'm around 850 fps? That is, BTW, my load for Hornady XTP 230grain hollow points. And I see it as an 850fps load, something that mimics the velocity that the original .45 ACP was designed to produce, even though none of the data points are exactly on 850fps, and there was an odd one that was low. The low data point doesn't invalidate the general trend, which is that a 5.7gr load of W231 at 1.225 oal can be expected to produce about 850 fps. I also evaluate the spread of the data, including the standard deviation. If I've got a lot of spread, then it suggests there is a lot of variability in my reloadingcould be the powder measure is dropping inconsistent loads, the bullets may not all be the same weights, may be seated to different depths, etc. I'm looking for standard deviation numbers in pistol that are no more than 2 percent of the average for a string of 10 shots, and less than 1 percent of the average for rifle. (I reuse brass for pistol, and it's not as consistent as I'd like, but the price is good.). If I get numbers like that, I consider the averages to be pretty stable. I think you may be placing too much emphasis on SD as a measure of quality, and ignoring accuracy in the process. Consistency is important, but so, too, is accuracy. 

April 12, 2010, 01:57 PM  #21 
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The statistician I used to work with always claimed it took about 30 samples to have what he considered reasonable confidence the bell curve was symmetrical, though under some circumstances he would use 21. I've forgotten when? SAAMI procedures calls for 10 samples in pressure testing to prove a prospective charge for production. It seems to work most of the time, though, once in awhile an overly warm load is approved and the lot is later recalled.
We don't know how many low pressure lots get through, since they don't recall those? I'd expect them to be greater in number because most commercial makers seem to have an internal extra safety margin they toss in; particularly for older chamberings. Military ammo is loaded to within both upper and lower performance and pressure limits, though, with the result you find more surplus military ammo that's inadvertently too hot than you do commercial ammunition. Obviously, if you fire enough rounds in load work up, that improves confidence your top load is not going to be producing unexpected extremes. I think that's what you were describing with your sets of 5? I like Dan Newberry's round robin approach for finding sweet spot loads, in part for that reason. You only fire 3 of each load level, but you've typically run through 2130 in total and at least 9 that hover immediately around any charge that's a candidate for selection.
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April 14, 2010, 10:12 AM  #22 
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A reloading objective is knowing that I'm satisfying the IDPA 45 ACP power factor requirement. Fortunately their testing for that requirement seems in light of the statistics we've been discussing to be quite generous. Three cartridges are selected and two of those have to make the power factor. Should that fail then three more are selected and tested. Two of three is just the minus 1 sigma level.
I'm getting better looking chrono numbers now presumably from culling out short cases, setting a slightly shorter OAL (1.250) and making a tighter crimp (Redding profile at 0.469). After these adjustments the same loadings show around 25 fps faster average velocities. Power factor 165,000 needs 718 fps for a 230g bullet. My latest WST test loading, 10 rounds, measured in at 759 mean, 37 ES, 14 SD or 1.84 CV (Coefficient of Variability? i.e. the SD expressed as a percentage of the mean). That would still make power factor down at the minus 3 sigma, 99% level. With apparently two SD's of PF cushion this load seems sufficiently calibrated. 
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