# Thread: Tech Tip #5 Statistics Without Math

1. AGD
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## Tech Tip #5 Statistics Without Math

Tech Tip #5 Statistics Without Math

In preparation for the analysis of the paintball breakage tests we have to cover the subject of statistics. Many people feel that statistics are unreliable and we have all heard that "you can lie with statistics" but good statistics tells us a lot. For those who are not mathematically inclined, don't worry there are no formulas in this Tech Tip!

We will examine how to measure the accuracy of a paintball marker even though it might not be very apparent. To do this we have to start with a typical shot grouping as you can see in #1 below.

The grouping in #1 is obviously not perfect and they never are, the shots are distributed in a wide area. If you wanted to measure the accuracy of this marker you could use a circular shape around the group as we see in #2 above. We all know that if you measure the diameter of this circular shape it gives you a number that you can use for accuracy. But wait, you say that there is that one "flier" at the top, should you include that? You might say "well that must have been a bad ball so we should leave it out". If we redraw the circle as in #3 our accuracy magically goes up!! Now we have a dilemma, which shape to use and what to include but it doesn't stop there.

If we look at figures #4 and #5 below we see that the same size circular shape can be misleading.

Figure #4 is obviously a tighter more accurate group with one flier, while #5 is worse overall but still fits in the same circle. So now we understand the dilemma, how do we measure these distributions accurately? Scientists face this all the time and fortunately there is a simple way to use another shape to more accurately tell us what we have. It takes a few more steps but they are not that hard.

The first step in statistically analyzing the distribution is to use more circles, as we see in #6 below, to give us some zones to count hits in.

Next in #7 we count the hits in each zone and stack them up. If a hit touches the line we will count it as being in the next zone. In #8 we mirror over the first three columns to the right side so our shape will be easier to see. This is the same data just copied to the other side for clarity.

Now we have a nice stack of hits and can remove the circles as we have done in #9 below.

In #10 we fit our new shape to the stack, look familiar? Sure it is, it's our old friend the Bell Curve!! If you ever suffered through high school math you must have run into this guy. I remember being "graded on a curve" aughh! So now in #11 we have a new shape, the Bell Curve, which has to be fitted to the stack by adjusting it's height and width etc. Programs like Excel do this for you today and just give you the results.

So now that we have our Bell Curve shape how the heck to we measure this weird thing? See #12 below!

Lets start at the top of the Bell Curve and slide down one side. As we just come off the top, the curve gets steeper and steeper. As we get about half way down the side the curve starts to flatten out. It is at this point, where the curve transitions from getting steeper to getting shallower, that we take our measurement.

We simply measure the width of the curve at that point. This measurement is called, are you ready, a standard deviation! Get it? A "standard way" to measure deviations! Just what we were looking for, yea! Much more sophisticated than a circle, way better than guessing and we got here without math. This measurement is also called one sigma and has it's own special symbol.

So what does the measure of standard deviation tell you? It means that 68% of the shots were inside this section of the curve. The dark section of #13 represents 68% of all the area under the Bell Curve. It tells you that this marker will put about 2/3rds of its shots in a circle one standard deviation wide. This is how the military measures their accuracy, this is how polls are determined etc. It allows for fliers but doesn't give them a lot of weight.

Now we are ready to compare two shot groups and see what comes out. Look at #14 and #15 below, which one has better accuracy?

In the two examples above, #14 didn't put that many right in the center but shows a little tighter overall pattern. #15 has a bunch dead center but a wider spread. Two shooters could easily argue that each had the better accuracy than the other. Note that they both fit in the same size circle so that doesn't help.

Below we see the results of a fit to each of the shot groups above.

We see the fruits of our labor as an obvious difference between the standard deviations of the two groups. If we were to take the group and input all the x and y coordinates for each shot into a program like Excel it would just kick out the standard deviation and we would be done. Doing it the long way gives a solid understanding of how we got the measurement.

The one thing we haven't covered is the error factor. In all statistics you get a result, plus or minus X amount. This is the amount that your result could be off if you did the same test again. You wouldn't expect it to be exactly the same but you wouldn't expect it to be way off either. We won't go into all of that here but you now know how to build the curve so errors will make some sense. In order to build an accurate Bell Curve you need a BUNCH of points! If you only had say 10 shots there would be no way you could fit the curve with any accuracy. If you had 1000 shots your curve could be fit very tightly. The moral of the story here is you can't determine accuracy from 5-10 shots. This is why I wanted 40 samples for the paintball test, otherwise the error bars would be so wide they would be meaningless. Sort of like firing one shot, hitting the middle of the target and claiming your marker is perfectly accurate!!

Statistics has shown us the light and made our job easier with less argument. We have learned that a circle is not the best shape to determine accuracy, the Bell Curve part of math really will help you with fun stuff like paintball and most importantly all of this is comprehensible to most anyone. I myself only have a high school math education so I might have missed some details. If I have, some smart people here on AO will be sure to catch it and correct me. If you want to impress on your parents that paintball is more than just a game, tell them in the forum tonight we were discussing a one sigma deviation of a standard distribution based on data taken from two shot groups. Mention you are worried that your error bars were getting out of control due to a lack of input but your Bell Curve should look better after more runs. That might net you some bucks for paint!

On another note Tom Kaye's Tech Tips will be a monthly feature in Action Pursuit Magazine starting in two months. We will be extending our search for truth, justice and the American Way to the rest of the paintball world!

Tom Kaye
Last edited by AGD; 12-16-2001 at 11:09 PM.

2. Kila Products
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Links/pics/diagrams not working. Cool tech tip though!!

3. Lucid Dreamer
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They're working for me. This is a very informative tip, there's a lot of Precalculus in there.

4. Grand Master
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argh... no more stats class please.
surprisingly enough everything made sense to me

5. Super Moderator
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Standard error is pretty revealing, but degree of confidence is also something to look at... which has do with the sample size needed so that you will not exceed a specified amount of error. It's kind of like how many shots you need to take until the curve starts to settle... gives clues to what a sufficient sample size is.
Last edited by Miscue; 12-16-2001 at 11:54 PM.

6. a stockaholic
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Okay Prof. Kay, when will the exam be? Reminds me of my second year at the U.

7. King of Polyester!
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WOWEE! Action Pursuit Magaziene? Thats Action Pursuit Games right? Anyways, that really cool! But how do I tell exactly how many hit dead center when the paint just runs all together? Maybe I'm stupid and missed something, because I'm still not 100% sure I understood it all(lol). But Yeah thats great!!

ENDO!

8. AGD
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Major,

You use a paper that punches holes and doesn't break the balls.

AGD

9. King of Polyester!
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OH! I see! I was imagining shooting at a paper setup on a hard surface. Thanx alot Tom!

ENDO!

10. INCEPTIONDESIGNS.COM
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A large corregated sheet of cardboard or an old box works very well also.

manike

11. Registered User
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Awsome Tech Tip Tom! Thanks! I think you beat Shartly for "Longest Post" award with that one!

12. Destructive Customs
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LOL
i'm sitting here studying for my stats final and see the title stats without math.............
Gecko

13. The Neighborhood Tech
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very cool. APG? very cool! I hope this will let more people in on how extremely cool AGD really is.

-Dave

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Good job Tom! Very interesting report. I give you an A, if you want to stay after school for some extra credit I'll bring it up to an A+.

But really, that was very interesting.

15. Slightly disturbed member
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## Re: Tech Tip #5 Statistics Without Math

Originally posted by AGD
On another note Tom Kaye's Tech Tips will be a monthly feature in Action Pursuit Magazine starting in two months. We will be extending our search for truth, justice and the American Way to the rest of the paintball world!
So now I have to hear all these guys with Spyders talking about bell curves now instead of their new Java-Tuesday-One-Shutter-Gun-o-the-Month? Fair trade...

16. Long time no see
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Holy flaming dung balls! My last exam on friday was on this very subject. Bell curves, confidence intervals, standard deviation wasnt included in here was it? But wow, I actually understand this mumbo jumbo.

Edit: NM, yes it indeed it was included somewhere in there.

17. Moderator? Mob Boss.
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Neat-o.

Major was gonna go try that on plywood lol.
*pop pop pop pop pop*
*thinking*
*hm*
*lets see that splt there looks like it was in the 8 range....*

Will the tech tips already posted on AO be put in APG first or will you start with fresh ones just for us?
We need an online tech class.. like.. hmm.. interactive.. where players can ask questions to The Man himself.
A interactive chat would be neat. Dunno how you could get that many ppl in a chat room.

18. JJ's Master
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Well, now I guess i'm going to have to break down and get a subscription to APG now.

Very informative.
Thanks Tom!!

19. Revered Sex God
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NNNOOOO!!!!!!

I took the AP Stat test 2 years ago. I got a 5. Within 10 minutes of walking out the door, I forgot all of the formulas, but now I'm trying to remember them, and taking out my old notes.... thanks alot...

20. The True Wang
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God I hate math!

21. Super Seizure Robot
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Great tip Tom! I went through the full gambit of Statistics course when I got my masters, and dam I have to say its one of the better introductions I have read on the subject matter.

Also as a side note, the bell shaped cure, ie Gauss's equation is on the German 5 DM note along with a picture of Gauss. You've got to love a country that puts scientists, engineers, and artist on their money.

22. RKL
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Element5 wrote

Also as a side note, the bell shaped cure, ie Gauss's equation is on the German 5 DM note along with a picture of Gauss. You've got to love a country that puts scientists, engineers, and artist on their money.
Better hurry if you want one. The Euro comes out on 01 Jan 2002.

23. Moderator? Mob Boss.
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They're switching all the way to Euro's? damn.
weell.. thats an easier trade to hard currency... if ya get my drift.. ahem.

24. Alcohol Fueled Brewtality
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There is one thing that can be misleading when translating the circular hit pattern to a bell curve.
Say out of 20 shots, every single ball hit low and to the left, say in the 4th ring out. Using this method you will actually get a double-hump curve. That curve obviously denotes inaccuracy, but on the target is would be the gun you would want above any other, having perfect accuracy. As long as you know the aiming correction you can compensate accordingly.
Just an observation, and there are probably a few more examples out there too.
Brent Jackson, PFB.

25. RKL
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Having re-read Tom's tip, I believe the term he should be using is precision ie the repeatability of an event.

After getting accuracy and precision reversed in my first interview after grad, my new boss had a simple explanation that stuck with me all this time.

Precision is repeatability. If you throw ten darts and they all land in the 4 segment, you are very precise.

If you have been aiming at the 20, you are not very accurate.

Having measured the precision of your shot pattern due to the barrel and paint combo, you need to adjust the aim point by the offset from original aim point to center of dispersion. I think Dury Dan did an article long ago on zeroing in.

Cheers

26. Registered User
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Originally posted by UTLadiesMan
NNNOOOO!!!!!!

I took the AP Stat test 2 years ago. I got a 5. Within 10 minutes of walking out the door, I forgot all of the formulas, but now I'm trying to remember them, and taking out my old notes.... thanks alot...
this maybe off subject.........but isnt a 5 the best you can do on an AP exam?

27. 1TimeAtBandCamp...TwEaK!!
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Well first off i'd like to say, this can be misleading, with paintball there are many more factors that come into play. The biggest ones being wind and gravity, since a paintball is a larger, gel filled ball, in comparison to a bullet (where this information comes from, to get the accuracy of a gun, like tom said this is what the military does), it is effected much more. Wind can change a paintballs course drastically, where as with a bullet its not as effected.

On a side note, MASSDRIVER - If you were using a rifle, I'd tell you to adjust your sight, in the world of paintball I wouldn't do this test aiming at a bullseye and testing your accuracy that way, i'd test the grouping of paint shot. Seeing how tight of a grouping you shoot would be easy to find with your example, just center the 'bullseye' of the circle chart to the center of your groupings, then run the test as already explained.

With me, I dont expect to hit the bulleye with my first shot in paintball, I expect to get a shot close THEN i adjust the gun in accordance of where my first shot went. Atleast thats when I play semiauto. So i'd say a tighter grouping would be the key factor there

28. Alcohol Fueled Brewtality
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Exactly Chaos.
My post was only to make those reading this thread aware you can hide things in a bell curve, using the above method, that would be readily apparent in the circular model. Also note that to make the bell curve work you do not use only one circle, but a pattern of circles so that you could grade accuracy statistically, which is really the only way to find out how accurate paintballs are, because of thier inherent inaccuracy.
As for "zeroing in", that is a different matter entirely.
It is common when testing a real rifle or handgun for accuracy to simply fire a string of shots with the gun locked down just to get the tightest grouping possible. Look in some of the gun magazines and you will see thier shots all over the target, but grouped tightly.
You can then adjust your sights or scope to get you as close as possible to the central aiming point.
Brent Jackson, PFB.

29. Super Moderator
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Bah. Ya guys are missing the point.

30. Alcohol Fueled Brewtality
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My first shot may have missed the point, but statistically I hit the mark.

Sometimes I just kill myself. hehehe
Brent Jackson, PFB.

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