Home Forums Hunting & Shooting Ask The Gunwriters Ken Howell - SD Questions from PS Article

Ken,

I just read your article on Standard Deviation in the Jul '04 Precision Shooting. I have always used the extreme velocity spread divided by two to get a feel for velocity predictability. I always thought that method would give me the worst case velocity spread.

Well, after reading your article, I have been thinking about applying SD and seeing what I get. Here are some results from yesterday's trip to the range with my 6mm Rem. Please tell me if I am applying the concepts correctly. here are the first six shots of this particular load.

3169
3154
3199
3215
3199
3184

AVG velocity: 3187 FPS
Extreme deviation: 61 FPS, or +/- 30.5 FPS
StdDev: 20.39 FPS

By using only the extreme velocity spread, it tells me average of future shots from this load in the same conditions should be within 30.5 fps of the average of 3187 FPS.

By using Std Dev with the first table, it tells me there is a 90% chance the average of future shots will be within 19 FPS of the 3187 fps average. I am using the following values for the table" 20 for SD and 5 for the rounds tested.

Here are the rest of the shots from that load fired yesterday:

3177
3147
3139
3147
3171
3171

The average of the last six shots is 3159 FPS. The average of all 12 shots is 3173 FPS.

So I guess this does verify the SD table, in that the average of all 12 shots (3173) was within 19 fps of the average of the first six (3184). Maybe SD is a more precise predictor of average velocities than extreme deviation.

Am I doing this correctly?

BTW, my best group (3-shots) went into .125" at 100 yds and chronoed: 3169, 3154, 3199; having an extreme dev of 30 fps. My worst group went into .625" and chronoed: 3147, 3171, 3171; for an extreme dev of 24.

I guess the math is useful here, but perhaps not so much for 100yd initial load development. I can tell by the groups and the the extreme velocity spread the load needs more work. I am thinking SDs for 6 shots should be around 5........

Comments?

Blaine

As much as I'd like to be able to, Blaine, I simply can not offer individual appraisal of shooters' loads and math. What you've presented here feels right enough to me, but I haven't the time to do all the math necessary to double-check the results of the tests that I'm already being asked to critique. Math is tedious for me -- checking my math even more so. I'm inclined to trust yours much more than I trust mine.

As I expected, and despite the fact that I sought the input of statistics experts on my article before I sent it in, I've already begun to hear critical comments about how I should've written it. I asked one critic whether I'd written anything that was wrong, and he immediately said "No." Then he proceeded to tell me what would've made the piece more palatable to statisticians. I reminded him that I'd (a) asked statistics experts for over thirty years and waited in vain for experts to write about standard deviation in terms digestible by shooters, (b) submitted drafts of my article for appraisal by statistics experts before I submitted it in final form for publication, and (c) warned readers of my own lack of expertise and fumbling on the subject.

If I understand his criticism correctly, the procedure that I described gives you only the SD of the test string, while he prefers to use a complicated calculus equation that predicts the SD of all the rounds yet to be fired. And if I understand his criticism correctly, the simple way to do this is to divide by n-1 instead of by n, as I wrote in my article (when n is the number of rounds fired in a test).

Ken,

I wasn't really expecting you to check my math, but to let me know if I have the concept down, which you have done--thanks!

I think I'd answer you critics in a couple ways. First, I'd point out you were only trying to provide the layman an introduction to SD and how to compute it. Second, I'd ask them where their article is on shooting and SDs, and point out Dave Brennan would be very willing to run their article too........That is, if they can figure out how to write it in an understandable manner...........

I'll bet I can work up a simple spreadsheet to calculate SD, where all you have to do is plug in the numbers. In fact, I bet others have already done this as well.

Blaine

Hey Ken,



I believe the statistician's preference for n-1 instead of n has to do with the idea of bias.

When you're doing these calculations you are estimating the value of the variance (or its

square root, the std. dev.) of a probability distribution from a sample data set. If the

expected value of an estimator equals the actual value of the parameter being estimated,

the estimator is said to be unbiased. It turns out that the division by n-1 stuff gives the

unbiased estimator for variance, and unbiased estimators are what statisticians generally

prefer. I'll leave it at that since I'm mathman, not statman, and more depth and detail would

be hard to get into in this forum. I hope this helps a little.



Dave



P.S. I should mention that getting an unbiased estimator of the standard deviation is not

quite as simple as taking the square root of an unbiased estimator of the variance. Also, to

any professional statisticians out there, I realize my post isn't exactly rigorous.

I was taught lo these many years ago that it has to do with confidence intervals based on the sample size, i.e. that if the sample size is considered great enough to generate, in general, +95% confidence then the general formula for developing a confidence interval for a population meanis used, where x is the mean of the sample; Z is the interval coefficient, which can be found from the normal distribution table; S is the standard deviation of the sample and n is the sample size. If the sample size is not of sufficient size to instill a high confidence level the �small sample procedure� formula is used substituting as the interval coefficient providing an area in the upper tail of a t distribution with n-1 degrees of freedom which can be found from a t distribution table.

BTW, Blaine, if you�ve loaded the �Data Analysis ToolPak� in Excel you can use the �Descriptive Statistics� tool that calculates summary statistics for a set of sample data including Mean, Standard Error, Median, Mode, Standard Deviation, Variance, Kurtosis, Skewness, Range, Minimum, Maximum, Sum, and Count just by highlighting your input range and selecting an output range.

And this would be why I studied journalism in college...

I am very glad there are people who can figure all this out, and I thank you for doing it.

One of my more dubious honors is teaching a course in introductory statistics at a local community college, and Journeyman has it right as to the proper way to relate a standard deviation from a sample to the overall mean (average) of the "population", that is, the average of all possible shots that might be fired with that load. The sample size (i.e., number of shots fired, in this case) needs to be 30 or more before you can reliably use the Z distribution. The small samples that handloaders usually use necessitates use of the t distribution.

By the way, the instructions to my old Oehler Model 33 chronopgraph (still in use) contained a condensed version of the t-distribution table and told in effecr how to set up the confidence interval that Journeyman mentions. It is this application that was the real intent of including the standard deviation on the chronograph's readout. Standard deviation is a measure of consistency, but not a very reliable one with the small sample size most handloaders (including this one) normally use. The trouble with the confidence interval is that it shows that we must accept a lot more spread in our velocities than we feel comfortable with, so, along with the fact that it's a little troublesome to compute, it's rarely used.

By the way, you can't differentiate between the standard deviation of a sample and that of the population just by choosing between n and n-1 as your divisor. It's got to do with "bias" as one poster mentioned (I won't attempt to explain further; it's one of those take-it-on-faith things). There is also a confidence interval you can do on sample variance, and its square root, sample standard deviation, which will give you a range for the population s.d. This isn't the format to investigate that, however.

BTW, Blaine has apparently used the "biased" formula (n divisor) to figure his s.d.; the more-appropriate "unbiased" version would be 22.3, which would spread his interval out a little.

Bob has hit the nail on the head, particularly with his mention of small sample sizes.

Although it has been many years since my formal instruction in statistics, as an engineer I use it nearly every day. I'd like to pass along a few observations.

First, there are so many stat books because no two statisticians can agree on the best way to do it. I've worked for several companies (currently in food, used to work in chemicals) and each one has applied statistics differently. Each employer has had a stat guru who has written his own text. The mentality goes something like this: "our product is unique, and thus demands a unique solution..."

Now, the mathematics involved is really quite straigtforward. It is what it is, and any decent introductory textbook will tell you what you need to know. The voodoo is in the interpretation of what you have calculated. Most of the time the results can lead you in more than one direction, so how do you make a conclusion? Large sample sizes really help. The larger the sample size, the less "interpretation" you will have to make.

Bob is entirely correct about the reliability of the Z-test. You need a sample size of at least 30. In my experience the Z-test is much more powerful than the T-test, so I'd encourage you to just shoot 30 rounds and use the Z distribution.

I realize that isn't always practical, but making sense of small sample sizes really is a black art. If you're interested in making it work, the best text I've seen is "Innovative Control Charting" by Stephen Wise and Doug Fair. This book deals specifically with how to make do with small sample sizes. It's pretty spendy, especially when you consider how thin it is, but it is really good. Quite a bit of advanced math, though, and not for everyone. Personally I find it less of a headache (and more fun) to just shoot a few more groups.

I don't have access to the table you mentioned, nor to the article. Can you explain again why you use half of the "extreme deviation" figure?

Using the definition of Standard Deviation I remember, you would apply the whole SD to the mean (average) to obtain confidence levels. One SD either side of the mean is one confidence level, two SDs either side is two confidence levels, and so forth.

With a mean of 3187, +/-20.3, therefore, you can be 66% certain (one confidence level) that your future velocities will fall between 3167 and 3207. You can be be 95% certain (two confidence levels) that your velocities will fall one more SD above and below those numbers, 3146 and 3228. You can be 99.4% certain (three confidence levels) that your velocities will fall between numbers one more SD above and below those, 3126 and 3248.

That seems to track, since all of your shots mostly fell in the 1 SD range.

What am I missing here?

Jaywalker

Using only the mean+standard deviation will describe precisely the sample that was used (i.e. number of shots in the string) and provide an estimate of the mean and standard deviation of the population (i.e. all of the shots that will ever be made with the load.) The weakness in this is of course that measuring another sample (string of shots) will almost certainly yield a different mean.



What you�re really after is "how good is the estimate of the mean?", not "how much variation was found around this one estimate of the mean?"

Journeyman: Granted, but isn't that always the issue - whether or not the sample is representative? If it isn't, then statistically, the sample isn't much use. If it is, the sample describes the future, also.

I still don't understand the use of a value that's half of the extreme - I'm not disputing it, just wondering what it represents.

Jaywalekr

Jaywalker,



I apologize for misreading your post. I thought you were questioning the value of performing the standard error/standard deviation of the mean and confidence interval of the mean calculations vs. mean +/- standard deviation.



I agree fully that the use of extreme spread or mean absolute deviation as an indicator of uniformity is at best dubious statistically. In fact, in my schooling the entire definition of �extreme� and how to address extreme data points was treated as a controversial subject since like the mean, one or two extreme scores, or �outliers� can broadly influence the standard deviation. I was instructed for obviously skewed data to use the semi-interquartile range (one half the difference between the 75th percentile and the 25th percentile) as it is little affected by extreme data points. Standard deviation is still preferred however as it is less subject to sampling fluctuations in a normal distribution.



Interestingly enough, after reading Bob�s post above I dug out the manual to my Oehler 35P and found this gem of a summary to this topic in general:

Taking multiples of standard deviation either way from the mean is just a description of normal or nearly-normal populations, and is not an analytical tool to use on samples, nor should it be confused with confidence intervals.

The level of confidence used in a confidence interval is arbitrarily selected by the user, and is typically 90 or 95%. The appropriate value of z or t is then used, together with the the formulas posted by Journeyman earlier.

For example, in small samples typically used by handloaders, for a 95% confidence level the appropriate value of t for a 5-shot sample is 2.776. If your 5 shots had a mean (average) of 3000 fps and a standard deviation of 20 fps, then your confidence interval would be from 3000 -(2.776)(20/square root of 5) to 3000 + (2.776)(20/sqrt(5)), or from 2975 to 3025. You can thus say with a 95% probability of being correct that the true mean of all possible shaots fired with that load (disregarding atmospheric extremes, etc.) woul lie somewhere between 2975 and 3025 fps.

If you wanted to narrow down your interval a bit, the way to do it is to take a larger sample, i.e., chronograph more shots. The values for t (at 95% confidence levels) for various number of shots is as follows:
No. of shots, n t
6 2.571
7 2.447
8 2.365
9 2.262
10 2.228
15 2.145
20 2.093

If you get really ambitious and shoot 30 shots or more, you can substitute z instead of t in the above procedure. The z value is independent of sample size (once it gets over 29) and is 1.96 for a 95% confidence level.

A secondary value of taking the larger sample sizes is that you will get more precise values of standard deviation.

Post deleted by bob_a

I apologize for the multiple posts--I kept getting a message that the post couldn't be sent for some reason, so I did it over, but it appears that it was going after all (it never did accept my editing, though-- one problem I tried to correct was the jammimg together of what was supposed to be two separate columns of numbers).

So delete two.