"Standard Deviation"?

[FAsmus]

Gentelmen;

I'm running into some problems in my arithmetic and, for back-up, I'm going to ask for help from the membership;

Could someone please post the formula for computing SD for me? My old one seems to be generating more trouble than anything else!

Good evening,
Forrest

[dagger dog]

Determine the average velocity of all shots,and find the difference between each shot and the average, each difference is squared and the total is added together, this total is then divided by the number of shots MINUS ONE,the square root of the answer is the standard deviation.

this info gleaned from UNDERSTANDING FIREARM BALLISTICS ,Basic to Advanced Ballistics,BY ROBERT A. RINKER.

Hope this helps

dd

[noylj]

Do you know what the S.D. and how to use it?
How To Calculate Standard Deviation
First, you need to determine the mean. The mean of a list of numbers is the sum of those numbers divided by the quantity of items in the list (read: add all the numbers up and divide by how many there are).

Then, subtract the mean from every number to get the list of deviations. Create a list of these numbers. You will get about half of the deviations as negative numbers. Next, square the resulting list of numbers (read: multiply them with themselves).

Add up all of the resulting squares to get their total sum. Divide your result by one less than the number of items in the list.

To get the standard deviation, just take the square root of the resulting number

your list of numbers: 1, 3, 4, 6, 9, 19

mean: (1+3+4+6+9+19) / 6 = 42 / 6 = 7

list of deviations: -6, -4, -3, -1, 2, 12

squares of deviations: 36, 16, 9, 1, 4, 144

sum of deviations: 36+16+9+1+4+144 = 210

divided by one less than the number of items in the list: 210 / 5 = 42

square root of this number: square root (42) = about 6.48

The only reason for the SD for most people who shoot is if they want to ensure that they will meet a certain power factor or if they have actually designed an experiment and want to determine the most significant variables and what changes will actually make a statistical difference. Very few actually even do the latter.

[Zbench]

SD = Square Root(1/(N-1))*Sum(X-Xbar)^2) is the most common one. It is used where you are calculating the SD of a sample from a larger population. N=sample size (ie, number of unique values in the calculation). X is an individual value, and Xbar is the average of all the values.

If you want just use a simpler formula which will calculate N for a population, you can simplify the N-1 term to just N.

Example: You shoot 4 shots and record velocities of 1000, 1100, 1050, and 900 fps.

N=4
Xbar = 1012.5

SD= square Root (1/(4-1))*Sum((((1000-1012.5)^2+(1100-1012.5)^2+(1050-1012.5)^2+(900-1012.5)^2)

SD=89.39

Excel does it nicely, but if you have to calculate it by hand, this does yield the same answer.

Pete

[FAsmus]

OK fellows ~ I have the procedures in hand, thanks to you.

Thanks very much, good evening,
Forrest

[c3d4b2]

If you have excel:

http://www.gifted.uconn.edu/siegle/research/Normal/stdexcel.htm

[Old Goat Keeper]

For less than $20 you can buy you a Casio solar calculator the will do average speed, standard deviation and extreme spread at the push of a couple of buttons. That is what I do.

Tom who used to work with statistics at work

[mike in co]

and the sample size is important.
while the formula will produce a "number", a significant number is based on a large sample size...something not done in the shooting world.
it is just a reference...and truely not mathimatically significant.

[geargnasher]

33 samples is a minimum for any meaningful results.

Geargnasher who once took a statistics class ONLY because it was an undergraduate requirement for engineers.

[runfiverun]

i push the button on my chrono.
it tells me all that.
worth the extra 20 bucks.

[scrapcan]

Ahh glad to see the sample adequacy statements fall into this discussion. Sample adequacy is a big deal and it is seldom done, especially in teh shooting world and even in some science fields.

You have the rest of what is need to calculate a the SD of a sample set as given above.

Rember numbers don't lie, but statisticians do!

[lwknight]

I second that runfiverun. I don't know what the big deal about SD is anyway.
Example : I had only Remington rifle primers and they are apparently weaker than winchester primers.
I used 54 grains 760 powder in a 30-06 with 150 grain bullets like the book said.
My velocities were terrible.
2600
3150
2434
2980
and so like that. I knew the load was no good and could not care less about the SD.

I upped the charge to 58 grains and guess what?
2990
2984
2993
2990
and so on. Then I started shooting one ragged hole also.
With those 2 things combined , I still couldn't care less about the SD.
Only the extremes mattered.

SD is really for demographics and legal matters.

[hiram]

Go to excel if you have it. Enter, let's say 6 numbers in column A. Move the cell marker(black frame) in cell A7.

1-Click on the large fx on the left side of the screen.
2-A window opens entitled 'most recently used'.
3-find STDEV on the list-- that's standard deviation. Click on OK.
4-Another window will open that says A1:A6 -- those are the six numbers you entered.
5-click OK and your stnd dev appears in A7.
Your done
The directions take longer to type than to do the entries. Its easy, it works.

[405]

I get tired of repetitive calculations.... heck I got tired of it in school and at work- even using a scientific calculator. Here's what I use, kind of a lazy way out.
http://www.easycalculation.com/statistics/standard-deviation.php

[Char-Gar]

What have iI been missing, 58 years of shooting and never once knew what the SD of my loads were? I thought where they landed on the target was the important thing.

[.45Cole]

SD is a stastical (yea I butchered that one) term. It is very good at giving a person a measure of precision of their data. If you counted peanuts in a jar but came up with a different number 8 times then you would calculate the SD of the counts. This would tell you if you are very good at counting, but calculating the error would also be beneficial to give you a measure of how accurate you count (different). For your loads, the SD will tell you how repeatable the load is (like how each shot is exactly like the others). It is worth while, but then again you have to want precision, and like counting peanuts and calculating the stastics (still not right). Most calculators will do this under the STAT function automatically if you put in the points. Or PM and I'll write a xcel file that calc's that stuff and email it to you.

[FAsmus]

c3d4b2;

Thanks for the link!

Wonderful stuff.

Good evening,
Forrest

[FAsmus]

lwknight;

Well, sure SD is the kind of a number that dims somewhat in practical use; as you point out: A good combination is a good combination.

In my shooting it is at least semi-useful since it enables me to determine a load's performance at short range prior to taking it out for the "real" shooting I usually do at fairly extended distances.

I have seen loads that didn't group all that well up close that still provided excellent SD and thereby did very well at things like 600 to 1000 yards.

Believe me when I say that you don't want to fiddle around with high/low problems way out there ~ there are way too many troubles associated with the regular problems of "condition"!

Good evening,
Forrest

[FAsmus]

405;

Thanks for the link, I'll give it a try very, very soon.

Good evening,
Forrest

[Tatume]

If you really need to calculate SD by hand, there is a shortcut formula, which is actually used by most calculators and computer programs, as it is computationally less intensive. Once you get used to it, it can save a significant amount of time and effort. It is the formula we used when we were graduate students and had to calculate SD over and over again as part of our assistantship duties (before the widespread availability of computers).

However, the shortcut formula is not tutorial in nature; that is, when you read the formula it does not explain what is going on. Therefore, I forbid my undergrad statistics students from using it. Besides, once they demonstrate that they understand the SD I allow them to use a computer program for all descriptive statistics.

The claim that a minimum of 33 data are needed for the SD to be meaningful is well intentioned and close, but not quite right. The minimum actually depends upon the actual variation of the data and the magnitude of the individual numbers. I once participated in a study of variation in 22 LR velocities for a well-funded precision rifle team. With a large data set assumed to be the population (collected in a NASA wind tunnel at night when the facility was shut down, using four chronographs simultaneously), I used a Monte Carlo model to calculate the SD of tens of thousands of samples drawn from the population, using various sample sizes. In this case, 20 data were sufficient to approximate the population SD with 95% confidence. If fewer data were used the results were unreliable and worthless.

There are ways to calculate an approximate minimum sample size required for a particular level of confidence in the SD. Usually this will entail a pilot study, but in the case of shooter statistics, we usually know the characteristics of the cartridge well enough to estimate the parameters.

SD is also proportional to the scale of the data. That is, an SD of 10 associated with a mean of 3000 indicates much less variation than an SD of 10 associated with a mean of 1000. When the data get larger, so do the SDs. You can easily convince yourself of this by generating a small data set, and another that is a multiple of the first. Say you have 20 numbers and calculate the SD = 2.3. Multiply each number by 100, and recalculate the SD. The new SD will be 230.

As several people have noted, the use of SD for estimating variation in ammunition velocity is almost always worthless because sample size is insufficient. There are other problems associated with the inappropriate use of SD. Unless one is using a sound experimental design, it would be better to avoid SD altogether.

Take care, Tom

[lwknight]

Everything that I can think of where SD is really useful leads to federal regulations , demographics and/or financial institutions in one way or another.

[splattersmith]

I found out if one just shoots one shot strings, the math is my kind of easy.[smilie=s:

[FAsmus]

Tom;

Thanks for your post ~ I believe I understood it fairly well.

Most striking was the (now obvious) factor that SD numbers are proportional to the scale of the data .. This is because the black powder boys are always bragging up their low SD numbers in some of the matches I attend: "My 45/70 always shoots single digit SDs!" they say, rather pridefully ~ and then ask what my smokeless load is doing.

Well now, my 44/63 will barely make the cut into single digits; something like 9.2 SD the last time I checked. ~ But then my velocity is going around 1330 ft/sec or slightly better as compared to their 1150 to 1200 or so.

Good afternoon,
Forrest

[405]

FASmus,
Well, the 9.2 SD is excellent by most measures. Sometimes low SDs reflect accuracy potential and sometimes they don't. But, for long range target shooting they become critical. It is simply another piece of information about a load. SDs, extreme spreads, average MVs, etc. are so very useful when comparing loads, working up loads with a new powder, working up loads with a new bullet and so on. Plus, some kind of good data that is recorded adds to the interest in the pursuit of accurate, reliable, safe shooting and reloading.

[MtGun44]

Without enough data, looking at SD is not only a waste of time, it is erroneous data and
provides no actual value. Ignore if under 20-30 data points are used, if you want to get
any benefit from it at all.

Bill

[cbrick]

Here is an old article written by Mr. Chronograph himself and his thoughts on Standard Deviation. I never reproduced this article on my web site because I never asked for permission to do so.


STANDARD DEVIATION
KEN OEHLER

The standard deviation is the number which describes uniformity. The smaller the number, the more uniform the velocity. A standard deviation of zero means every velocity was the same. A standard deviation of 32 means that 2/3 of the individual velocities should be within 32 fps of the average.

The secret of making smaller groups is uniformity. Other things being equal, the more uniform you can make the ammunition, the more likely it will shoot to the same hole. Uniform velocities are simply another indicator of uniform ammunition. Uniform velocities do not guarantee small groups, nor do large variations guarantee large groups. There are no guarantees but you can put the odds on your side. When velocities are uniform, you can assume you have a proper primer for the powder, that you have a reasonable powder for the case and the bullet, that you did a good job of measuring the powder and that your cases were of uniform capacity. Uniform velocities tell you very little about bullet quality, the bedding of the action and barrel or if the gun vibrations induced by the firing just happen to fall in a sweet spot. When erratic velocities and small groups predominate, your bedding is probably good and you have a good average velocity for that powder/bullet combination, but be suspicious of your primer choice and firing pin. When you experience both erratic velocities and large groups, go ahead and make significant changes in bullet, powder or gun; you probably aren’t close to any perfect combination.

Theoretically, you could use the standard deviation as the only measure of the uniformity of the ammo. The common limitation on the formal use of standard deviation and other statistical procedures in shooting is the number of shots required. Statisticians call it sample size. Invariably, statisticians want to see at least 20 shot samples. Nobody questions that firing more shots into a group will give a better statistical measure of the accuracy and the standard deviation but it’s expensive and time consuming.

Quite bluntly, trying to measure the velocity uniformity of your ammo by chrono-graphing only five shots is in the same league as determining it’s accuracy with a single 5 shot group. A solitary 5 shot group is an indicator, but you can’t guarantee it will repeat itself. Likewise, a standard deviation number should be considered only as an indication of uniformity. Although standard deviation is the best available measure of velocity uniformity, it is still not good enough to be considered the overall measure of ammo uniformity. Use the standard deviation numbers as indicators of uniformity but use them along with other indicators of a loads performance.

Comparing “good values for standard deviation” is like giving someone a “good group size” or a “good average velocity”. It all depends on what your trying to do. In all probability you will use the numbers only for comparison, and you don’t compare apples and oranges. If your working up a silhouette load for a 44 mag, any comparison with a 45 ACP is irrelevant. You don’t compare the average velocity of a 44 mag with the average velocity of a 45 ACP. You don’t compare the groups from a 22 PPC with the groups from a 375 Elk rifle. You don’t compare the standard deviation from a good silhouette gun with the standard deviation of a 300 H&H mag. The only comparisons that matter are those made between the 300 H&H loads you decide to keep and the 300 H&H loads you decide to abandon.

What do you do with the group sizes, average velocities and standard deviations reported by another shooter with another gun similar to yours? They can influence which loads you select for trial. Choosing a load which worked for him will probably beat a random load from a manual. After you have tested the other shooters load, his results shouldn’t sway your decision. What really counts is how the load performed in your gun, when compared to other loads in your gun.

We agree that velocity uniformity is not adequate as the only criteria for selecting a load but it should be considered, along with everything else you can learn, by testing the load. One precaution is that the measured standard deviation includes variations in both the ammo and the chronograph. Chronograph systems with inadequate spacings between skyscreens often give passable readings of average velocity but questionable readings of standard deviation.

Whenever you use standard deviation, remember there is an important corollary of Murphy’s Law.

Large groups usually repeat.
Large groups with large standard deviation always repeat.
Small groups caused by luck never repeat.

[legi0n]

google docs has a pretty nice (and free) spreadsheet which can compute the SD
you just have to sign up at docs.google.com

[FAsmus]

Cbrick;

Thanks for sharing all that wisdom.

For some time I have suspected my chronograph's consistency; For example, it seems strange that the same load, the same day, the same rifle, only a different battery in the timer would cause a 20 ft/sec difference in clocked velocities. As a result your remarks regarding this subject hit home.

Without going into a laboratory style chronograph would you care to recommend a particular make & model?

Good evening,
Forrest

[cbrick]

Forrest, I can't be much help in recommending a chronograph in todays world. I am still using Dr. Oehler's chrono from the mid 70's and they haven't been made for several years now.

If for some reason I had to replace it I would be in the same boat as you, I would have to do some research on it. I talked with Dr. Oehler at the 2010 SHOT Show last January and he hinted that it may be possible that Oehler may re-introduce a new model to the handloading market sometime in the future. Sadly I've heard nothing of this since.

Rick

[FAsmus]

Rick;

Roger that.

I may go looking around a bit but, really, I don't do all that much load development from the bench these days ~ once a combination shows promise I take it out for long range proving; if there are any significant high/low fliers with good holding the load goes back into the development stage.

In this particular series of shots (50) I was trying to see if fillered loads would show; #1 Better SDs than non-filler while; #2 Group size showed any interesting differences between the two.

The test floundered when the sky got so bright that I began losing times, even though I had the shades over the photo cells. This handicapped the results' validity but from I did get it seemed that non-filler was better in SD and in group.

Good evening,
Forrest