I seldom set up to test loads without the pact chrono set up also. If I figure I don't have time to mess with it, I don't go shooting at all.
It's a different story if I'm just going to do some fun shooting, plinking with established loads.
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I seldom set up to test loads without the pact chrono set up also. If I figure I don't have time to mess with it, I don't go shooting at all.
It's a different story if I'm just going to do some fun shooting, plinking with established loads.
The chronograph calculates the average velocity of the sample of bullet velocities measured.
Bullet velocities vary, and two measures of this variation are commonly calculated by the chronograph. These measures are "extreme spread" or ES, and "standard deviation" or SD. (Statisticians call ES the "range".)
So, for example, we might shoot a sample of 10 bullets over a chronograph, and see that the average velocity was 1506 feet per second (fps), the ES was 62.4 fps, and the SD was 20 fps.
SD has the characteristic that its value is independent of sample size. If we shoot sets of 4 shots with a given load, or sets of 24 shots with that same load, the SD will average the same. Velocity SD, for any given load and set of conditions, is a constant.
ES for that same given load and set of conditions varies with sample size, the number of shots fired and chronographed. ES is mathematically connected to SD so that if the sample size and ES are known, SD is easily calculated.
With 10 shots fired, ES is 3.08 times SD; keeping in mind that SD is a constant. So in the example above with SD of 20 fps, ES should be 3.08 times 20 fps or 62.4 fps; vs. the example 62.4 fps. (For 10 shots, SD times 3.08 will AVERAGE about the ES, individual samples will vary in the relationship.)
The relationship works the other way, of course. For 10 chronographed shots and ES of 62 fps, SD = ES/3.08 or 62/3.08 ~3.08.
Here's a table of the sample size/corrections from SD to ES and back:
2/1.13 Sample size of 2, SD times 1.13 = ES, ES/1.13 = SD
3/1.69
4/2.06
5/2.33
6/2.54
7/2.79
8/2.85
9/2.97
10/3.08
15/3.47
20/3.73
30/4.08
40/4.32
50/4.50
75/4.80
100/5.03
When we talk about the SD and ES of velocity of a given load, we're talking about the same variation with numbers that are a multiplier/divider apart. They're the same thing, as long as the sample size is the same.
If the example was changed to reduce the sample size to 5, knowing that the SD will remain constant at 20 fps, we would expect the ES to be 20 X 2.33 = 46.6 fps. The ES falls from 62.4 fps to 46.6 fps. ES varies with sample size, so it clearly doesn't make any sense to talk about ES without specifying sample size. ES without sample size is meaningless, SD is constant and independent of sample size.
If we chronograph 6 shots with Remington 2 1/2 primers and 15 shots with WLR primers-everything else the same- we might get an average velocity with both loads around 1400 fps; ES with the 6 shots of 76.2 fps and ES with the 15 shots of 104.1 fps. Do the WLR primers cause more velocity variation than the Remington 2 1/2s? No, because if we divide 76.2 by the 6 shot multiplier/divider of 2.54 we get SD = 30 fps; and if we divide 104.2 by the 3.47 we also get SD of 30 fps. We didn't see a change in variation, we saw a change is sample size.
Remember, these relationships hold on the AVERAGE, over lots of samples. Your individual results will vary, but the principles hold true.
joe b.
The SD remains constant (% wise) only if the samples ingnition is uniform. If all 5 shots of the small sample have uniform ignition the SD will be different from one where 1 or two shots (same load) have poor ignition. Also numerous samples (lets say 5 five shot strings of the same load) more than likely will have slightly different average velocities along with slightly different SDs. The SD is an important measurement of uniformity but remember that a SD 14 fps only means that 2/3s of the shots will be within 14 fps of the average (mean) velocity. Question then arrises; what about the other 1/3 of the shots that are outside of the 14 fps SD? This is why we must also consider the ES when evaluating loads. Especially those used at longer ranges.
To Quote Dr. Oehler; "When ever you use standard deviation, remember there is an important corollary of Murphy's Law. Its regular use can replace many mathematical theorems and complicated statistical procedures.
Large groups usually repeat.
Large groups with large standard deviations always repeat.
Small groups caused by luck never repeat.
And so I've found, especially with cast bullets.
Larry Gibson
Larry Gibson's post mirror's my experience fairly closely. I too have been chronographing since the 70's and have kept detailed notes on all load devolpment. Notes include details from powder lot numbers, temperature and humidity, date, groups, alloy, BHN and more. A prime value of the chrono AND the notes is in comparison of your loads in your guns. What did this change do in this gun? What does this powder do at that temperature. Does a BHN change effect groups or velocity? Primer change effects and much, much more. The more load comparisons you can make the higher the value of all of the notes and all of the previous testing. It builds on itself and becomes more and more useful and pertinent to your loads in your guns.
In the early 90's I put all my notes in the computer and used the computer exclusively ever since. Live and learn I guess, after a couple of moves and not giving much thought to the paper notes of the past I no longer had them. One hard drive crash and I no longer had any notes from about 20 years of testing. Notes today are kept on two computers AND a paper notebook.
Another prime benefit of the chrono in working up loads is a pressure indication. NO, the chrono will not tell you what the pressure is BUT in mid powder capacity cartridges (mostly what I work with, say 7 TCU up through 308 Win) an increase of 0.5 gr powder will net right at 50 fps velocity gain. It may be 40 fps or 65 fps but it will be right around 50 fps. The first 0.5 gr charge increase that does NOT net an increase of the average 50 fps and you have reached the pressure max for that cartridge, powder, bullet combination. From that point on the only thing that can be gained is more pressure with NO velocity gains and very possibly, even probably velocity and accuracy loss. Its the point I back off a half grain.
Rick
Last edited by cbrick; 03-04-2008 at 03:29 PM.
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I'll second this.Originally Posted by mike in co
I used to chrono my reloads when shooting IPSC to see where I was on the "major/minor" power scale. It was the chronograph that told me my +p+ LE-issue 124 grain Hydrashoks were giving me almost .357 Magnum velocity out of the gun I was carrying at the time. I felt better about carrying a 9mm after that, because with a .357 Magnum, I fear neither man nor beast.
But it was during my initial years in reloading that I really enjoyed the chronograph. Enjoyed seeing what the powder people guestimated for fps versus what my guns and loads actually shot. Each gun and each barrell shoot different. I like knowing what's going on.
'Course, my chronograph went belly-up some years back and I haven't replaced it . . . yet.
Jeff
If livin' in the dark is what a man is after by all means avoid it. But IMHO I doubt that any one on this site can't afford a chrono, I know that most of us have spent more on one mold.
joe,
That just ain’t so, SD calculation uses a complicated formula.
Standard deviation is the ”root mean square deviation of values from their arithmetic mean”. To find the SD of just 2 values requires six arithmetic calculations. None of those calculations use the ES of the values.
1- find the arithmetic mean of the values
2- find the deviation of each number from the mean
3- square each of the deviations
4- sum the obtained squares
5- divide the sum by the number of values
6- take the non-negative square root of the quotient =SD
Microsoft Excel has a Standard Deviation function, =STDEV(cell range) and I used it to calculate SD for two strings of 10 numbers each, both with an ES of 100. Using your formula (ES/3.08), 100/3.08= SD 32.46, but in neither case did that work out to be true.
example 1
100
150
150
150
150
150
150
150
150
200
ES=100 - SD=23.57
example 2
100
100
100
100
150
150
200
200
200
200
ES=100 - SD=47.14
SD is not constant and has very little to do with ES. SD is a mathematical formula that gives “probability” (or best guess) of result based on a sample. I’m with Larry on this one, SD is all but meaningless without knowing the actual ES of the sample load.
joe - What your formula(s) does (basically) is take the Highest Possible and Lowest Possible SDs for a given sample range value and average the two. That defeats what SD is all about.
Last edited by Whitespider; 03-04-2008 at 07:37 AM.
Whitespider;
This ain't a stat class, but I'll try to help you out here.
The SD of velocities for any given load/gun combination is what it is, a constant, until maybe the gun barrel wears. Here we're talking about the population or process SD. This area gets sort of involved and opinionated.
Means if we measured a lot of velocities we'd get a mean and SD describing the population.
Now, sample SDs calculated from samples of size "N", will vary. So, if we shoot sets of say 5 shots and calculate the SDs, those SDs will vary. Say we shoot a lot of sets of 5 and chronograph them and calculate the SDs. The SDs will cluster around and average the population SD. There is no difference between the AVERAGE sample SD and the population SD. NOTE--The sample SD is an UNBIASED ESTIMATOR of the population SD.
Now, using the velocities for the same sets of 5 shots, calculate the ES, called by those statisticians the "RANGE". The ES values will cluster around a mean or average, but that mean WON'T be the population SD. it will be 2.33 times the population SD. So, we say that for sample size of 5, the ES = 2.33 times the POPULATION SD, or through the magic of algebra, that the POPULATION SD equals ES/2.33. For N = 5. ES is a BIASED estimator of SD, with N =5 the bias is 2.33.
Sample ES and SD are estimators of POPULATION SD. For any given sample these two estimators may not give the same estimate of POPULATION SD, but over the average of many samples they will.
Remember, if we shoot and chronograph 15, for example, shots, and calculate the sample SD and ES, unbias the ES estimator of POP SD, and see that they are different-we don't know which is closer to the POP SD.
The ES has been used as an estimator of SD for many years, particularly in Statistical Quality Control where it made statistics possible on the shop floor before the computer.
A computer search on range and SD will provide a wealth of information.
Also see "Tables of Range and Studentized Range", H. Leon Harter, "The Annals Of Mathematical Statistics", December 1960.
It's interesting to see your "root mean square" definition of SD, I use that definition or approach and seldom see others use it. The AC voltage in the wall is a RMS value, as is the measure of surface finish made with a profilometer, RMS microinches.
So, I hope this helps you out. I'd be happy to send you the 1960 article with tables that I find very helpful.
Good luck;
joe b.
Larry;
You're confusing the SD of a SAMPLE-which is what you and all of us see calculated on the chronograph, and the SD of the POPULATION. Sample SD and unbiased sample ES are both estimators of population SD. We can only ESTIMATE population SD, can't measure it.
ES is ALWAYS greater than Sample SD.
Velocity distributions with all components held as constant as possible- means same powder, charge, bullet, lube etc-same everything within the bounds of the possible, as we should experiment-will be A distribution with A SD, SD is constant for the POPULATION.
I think you need to unconfuse the population and sample and their SDs.
Remember, ES with varying Ns is a measure of N, not velocity spread.
joe b.
I'm on my 3rd. Chronograph. I use the same techique as cbrick, and it works well. As to records, I keep detailed records with targets. Hot loads worked up in early morning temps sometimes turn into mightmares at mid afternoon. Helps you find that the rifle has a sweet spot at what velocity to give you consistant small groups.
Joe
I'm not confusing anything. SD is not a constant. Perhaps you should further research that. SD is derived from from the 6 calculations that Whitespider discusses. As you see from his example the SD can be very different from the sets of figures with the same ES.
You are correct that what we get is a "sample". However the larger the sample the more relevent it is to the "population". The only way to know the exact ES, SD and mean velocity of a "population" is to shoot the entire population. Not really a practical thing to do. However getting a good "sample" gives you a reasonable expectation (or educated guess) of how the "population" will perform.
For example; I get a good ES with attendant SD of 30-45% of the ES from a 5 shot string of a test load. I load up 3 ten shot strings or 1 twenty shot string or 5 five shot test strings to test. All are good "samples". When fired if these test strings also demonstrate the same good ES and SD as the original 5 shot string then we have a good indication that the "population" will also perform as well. The ES and SD are not expected to be exactly the same but are expected to fall within the expected parameters of a "good load". I can then expect that additional rounds of this exact load using the same componants (the rest of the "population") and shooting them out of the same firearm will give close to the same ES and SD of the "samples" tested. In other words I can expect the rest of the rounds loaded to fall within the ES and be close to the average (mean) velocity. Yes that is an "estimation" but in my opinion it is then a pretty educated one.
If we take Whitespiders two examples I would bet the load with the SD of 47 would shoot a more consistant group than the example with the SD of 23. The 23 SD group would no doubt have 6-7 shots in a smaller group than the 47 SD group. It would also have 3-4 shots that were "flyers"; definately shots out of the group. The size of the groups may be the same but the 47 SD group would be consistantly more accurate to shoot. The reason is if we zero the 47 SD to the center of the group and the 23 SD to the center of the 6-7 shot group (where the SD tells us it is "good") we will find with the 47 SD group no shot is more than 1/2 the group size away from the point of aim. Conversly with the 23 SD group the 6-7 shots are no more than 1/4 of the way from the point of aim but the 3-4 "flyer" shots will be 3/4+ the distance from point of aim.
Fact is you will shoot worse with the load with the smaller SD in this case. That is why the two, ES and SD, must be considered together and the % relation of the SD to the ES must be correct.
Larry Gibson
Last edited by Larry Gibson; 03-04-2008 at 12:36 PM.
You have to take the distance of each shot about the center of the target (POA), and then correlate that with the muzzle velocity ES of those same shots, and then repeat the same at a different yardage. Now correlate the whole shootin' match as a composite to get any real meaning of the math. Besides, because we are talking vectors here, it would make good sense to use a cubic function for deviants rather than using a quadradic function. Numerical signs and angles are significant for this logic, and must be taken into account. All of this would measure the capability of the system to place shots consistently. Hint: Taylor series solves many complex arrangements fairly easily. We're getting out of the realm of a hobby. ... felix
felix
As I said, this isn't a stat class. You can believe and do whatever you wish, opinion triumphs over fact, as usual.
joe b.
Joe,
FACT; Standard Deviation is the “root mean square deviation of values from their arithmetic mean”.
FACT; Six arithmetic calculations are used to find Standard Deviation.
FACT; Standard Deviation is most accurate when the “arithmetic mean” is equal to the “arithmetic median” of the sample (a condition known as “normal distribution”).
FACT; Samples (and populations) of firearms ammunition with HIGH ESs typically have “arithmetic mean” and “arithmetic median” some distance apart from each other (rendering the SD less accurate).
FACT; I did some research Joe; your formula is not intended to calculate actual SDs or ESs. Your table of corrections is used to ESTIMATE Standard Deviation when only the “Sample Size” and “Extreme Spread” are known (actual values of individual samples are unknown), AND the distribution is assumed to be “normal” (mean and median equal or relatively close).
FACT; I’ve checked your formula against a bunch of my records; IT JUST AIN’T SO! Ammunition rarely has “normal distribution”, if it did we wouldn’t need a chronograph to do anything except show us velocity.
If your formula was correct for determining population SD with ammunition, the larger samples should be closer to your formula. In fact, the larger my recorded samples, the farther away the recorded SDs are from your formula.
Yes, I use my Pact on every session for working up loads for both rifles and pistols. Either that or your just guessing what velocity your getting?
"The Original Point and Click Interface was a Smith & Wesson."
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Again, Joe, statistics is only an ATTEMPT to explain the FACTs of realism! Because statistics is a man-made set of rules, you can exactly define these rules to be FACT! These facts are those being discussed by Whitespider and they are NOT the facts of the natural system we live by. So, please, don't insinuate opinions are on the same plane as facts; statistical calculations are nothing but a collection of opinions about facts. There is no way any kind of opinion can overtake fact. ... felix
felix
Joe
I did not derive the mathematical equations to calculate SD nor did I make the definition of SD. SD is a standard set of equations that are accepted as fact by far more learned men than you or I. Those equations are also the standard fact with which the arms/ammuntion industry, chronographs and computors (also those who wish to do the tedious long hand) determine SDs. Unless we use the accepted standard, or fact if you will, we will be comparing apples to oranges if we use a different method of calculation. That is fact, not my opinion.
Larry Gibson
IIRC, back when I took a statistics class the book claimed that the sample standard deviation was an estimate of the population standard deviation but it had been determined that the sample SD converged with the population SD if the sample size was 30 points or more. FWIW, in SPC I always used a minimum sample size of 30 pieces for the initial number crunching and used +/- 4 standard deviations to estimate the E.S. for products.
Rick
____________________________
If it looks plumbous, I'll probably try making bullets out of it. Dean Grennell
Whitespider;
Maybe this is a stat class. I'll try to help you guys out.
"FACT; Standard Deviation is the “root mean square deviation of values from their arithmetic mean”."
The population SD = RMS deviation, true. However, the SD is also some other things. Normaldistributions have inflection points at the SDs, and there are are a lot of SDs.There's 1 SD above the mean, 2 SDs above the mean, 4 Sds below the mean, etc. At each there's an inflection point, where the second derivitive goes to zero.
The SD, along with the (arithmetic) mean are the defining variables in the definition of (formula for) the normal distribution.
"FACT; Six arithmetic calculations are used to find Standard Deviation."
Well, not true, not a fact. RMS deviation = population SD, but I can't imagine how or when we might try to calculate a population SD. A FACT is that SAMPLE RMS deviation isn't, I don't think, a good estimator of POPULATION SD. Give it a try, against EXCEL calculations or your own. Now why is that…….?(A prize if you can tell me!)
"FACT; Standard Deviation is most accurate when the “arithmetic mean” is equal to the “arithmetic median” of the sample (a condition known as “normal distribution”)."
I don't know what "most accurate" means. You're confused here, again. The median is another "mean" or "average" or indicator of central tendency. The median value of a set of data or a population is that value which there are as many values above as below. In the set 1,2,3,4,5; 3 is the median-there are two values, 1 and 2, below it, and two values, 4 and 5 above it. (There's also the mode, the most frequntly occurring value. And many others.) Now your logical error. All normal distributions have the median = the arithmetic mean or what we call the "average"; but not all distributions with arithmetic mean = median are normal. Consider the distribution of age of 70 year old men. Their average and median age is 70 years, but the distribution of age is not normal. No, it isn't. How about the age of first graders?
"FACT; Samples (and populations) of firearms ammunition with HIGH ESs typically have “arithmetic mean” and “arithmetic median” some distance apart from each other (rendering the SD less accurate).
I don't know that this is so. I think that you're saying that gun velocities aren't distributed normal. If you have some data, I'd like to see it.
Again, "SD less accurate", what does it mean.
We assume that many variables are distributed normal, so that analysis can be done. Many times we don't KNOW that the variable is distributed normal. I would suspect that gun velocities ARE distributed normal, don't know.
This would be interesting to look at. How about sending that data?
"FACT; I did some research Joe; your formula is not intended to calculate actual SDs or ESs. Your table of corrections is used to ESTIMATE Standard Deviation when only the “Sample Size” and “Extreme Spread” are known (actual values of individual samples are unknown), AND the distribution is assumed to be “normal” (mean and median equal or relatively close).+
I don't have a formula. I attempted to tell Larry that ES and SD are related, and that ES is a function of both SD and N, the sample size. The-not my-table of corrections is or may be used to estimate SD from ES when N and ES are known. We can also know the values of the sample variables. Whether or not we know the values of the sample measurements, we can still estimate SD from the combination of ES and N. Again, we assume a normal distribution.
"FACT; I’ve checked your formula against a bunch of my records; IT JUST AIN’T SO! Ammunition rarely has “normal distribution”, if it did we wouldn’t need a chronograph to do anything except show us velocity."
Formula? Please send or post data showing that velocity isn't distributed normal.
"If your formula was correct for determining population SD with ammunition, the larger samples should be closer to your formula. In fact, the larger my recorded samples, the farther away the recorded SDs are from your formula."
See above.
I'll go through some chronograph data and post the results here.
Joe b.
OK Felix, you're next;
Statistics isn't "an attempt", it isn't a "man-made set of rules", statistical calculations ere not "nothing but a collection of opinions about facts". Statistics isn't an opinion.
Statistics is a branch of mathematics, a mathematical topic, a mathematics division, a part of the greater "mathematics".
I told my students that defining statistics or any subject of study such as, for example, geology or religion or sociology, is hard to do in a few words. But, I told them that a short definition of statistics that I like is "a part of mathematics that translates numerical information into statements in words". So, statistics allows us to make statements such as:"Based on a sample of 456 persons polled, we find that 57% of citizens prefer chunky peanut butter to smooth peanut butter. We're 95% sure that the 57% is within +/- 3%, means we're 95% sure that 54% to 60% of the citizens.........."
Now, there is a fact that the ES and SD of a set of data is related, that SD is a function of ES and N, and it is a fact that there are folks offering opinions about statistics and this relationship. As you know, Felix, it is improper to offer opinions about facts. Opinions don't have any place in discussions of the value of pi or e or the atomic weight of zirconium.
The facts of realism? Felix, what can that mean? If you have something to say, how about short declarative sentences? Otherwise, I begin to wonder...........
joe b.
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