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Some math and stats for Sav17
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Old 07-05-2005, 04:05 PM
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Default Some math and stats for Sav17

Others read at their own discretion - it might get a bit 'tedious'.



Sav17 asked for an explanation some of my math, so I will explain it. You will find clarifying parenthetic comments in orange, and items needing emphasis inyellow. You can skip over the orange for readability, but you should pay special attention to those words in yellow. Since I cannot see Sav17's face, (or 'yours'), I won't know if I'm being too detailed or not, so please bear with me. I'm not talking 'down' to anyone, I just want to be 'thorough'.



I'll start where most folks start when 'getting to know' a new rifle - accuracy. Actually, since this is a thread about the math and statistics of ballistics, let's use the technically correct term - precision. And, for the same reason, let me differentiate between accuracy and precision. Accurate is how close a shot or a group of shots is to the exact Point of Aim (hereafter PoA). (Accuracy is what target shooters ultimately must strive for.) Precision is how small a group of shots is, regardless of where the PoA was. Precision is where everyone must start, and usually where most hunters remain, by virtue of the fact that most of them end up having their Point of Impact (PoI) one to three inches high at 100 yards. Technically, very inaccurate in fact.



Here's an example. Let's say two riflemen show up at the range together. One's rifle shoots a half-inch 3-shot group that's 2" high at 100 yards. The other shoots a 1" 3-shot group whose center (average PoI) is exactly the center of the bull’s-eye (PoA). The first shooter has the more precise rifle, while the second shooter and rifle are more accurate. A rifle alone cannot be accurate - it can only be precise. Only a rifle AND a shooter can be accurate or accurate AND precise. Put another way, accuracy is solely the responsibility of the shooter, precision is a function of the rifle AND the shooter. Gunwriters have mis-used these terms for so long, that "accuracy" is now synonymous with "precision" in casual conversation. It usually is only among those that really care about correct terminology that the term "precision" crops up in conversation.



So... Most folks are 'concerned' about precision when they bring a new rifle home. The way most measure, (actually estimate is the correct term), that precision is by shooting 'groups' at some fixed distance. Usually it's 3 or 5-shot groups at either 50 or 100 yds. (Since Sav17 is English, and most of the members of THL are American, I am going to stick to yards, feet and inches, instead of meters, joules and centimeters. No 'arrogance', just keeps the focus on the subject, not the language.) The distance between the most distant two bullet holes is measured, and that value is reported as the 'accuracy'.



There's a VERY IMPORTANT CONCEPT HERE - PREDICTION.



Regardless of how much bragging goes on, the real reason for performing such an exercise is to predict where the NEXT shot is going to go.



If you are a hunter, you want to have some idea about where a bullet is going to go when you pull the trigger. All this range work is ultimately about predicting (being 'sure') that the bullet is going to go where you aim it.



With the significance of "prediction" noted, I think it can reasonably be said that improving one's ability to predict has value, especially if that improvement is 'easily' obtained. Seeing as physics, math and statistics are part of my job, it was easy for me to see how I could increase my predictive capabilites using exactly the same basic data, and simply doing a little extra math and record-keeping.



The 3-shot groups represent a sample from a 'normal' (a poor choice of words used primarily by the biological community to describe the statistical "bell curve". Physicists, engineers and mathemetician usually use the term "Gaussian distribution", thereby avoiding any confusion about what is "normal"), distribution of a random variable. The random variable we're focused on is Point of Impact. Since we have what is assumed to be a "normally" distributed random variable, (I'll get back to what happens when the variable has statistical "bias"), using the "x" and "y" coordiates for each bullet-hole, we can calculate the average PoI, and more importantly, the variation around that average. That variation allows us to make much stronger predictions about the precision of our rifle. In other words, we can make better predictions. Furthermore, by keeping reasonable records, we can pool the data for our rifle over years, and see how its performance changes with time or other variables. We can also determine if there is a 'trend' (a 'bias') in the data. Trends usually don't show up with small (3 & 5-shot groups for example), sample sizes. This will be clearer when I discuss specific examples and use pictures.



So... let's get to some specifics. Let’s use Sav17’s targets for examples when possible. The first image below is one Sav17 posted of one of the first targets he shot with his .243. Following it are the derived "computer targets". The first is the first three shots in the circle labeled “Sc Adj 3” which I assume to mean “Scope Adjustment 3”. The second generated target is for those shots contained in the circle labeled “Scope Adjust 4”. The third generated target is for the combined six shots with the effect of the ‘scope adjustments taken into account. There’s great stuff here, and I’ll discuss it in the next post.

Paul
Attached Images
File Type: jpg Richie target243.jpg (32.9 KB, 73 views)
File Type: jpg Sav17 .243 003.jpg (32.7 KB, 78 views)
File Type: jpg Sav17 .243 001.jpg (31.4 KB, 84 views)
File Type: jpg Sav17 .243 002.jpg (31.5 KB, 91 views)
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Tags
95% ci, accuracy, figure of merit, fom, math, maximum point blank range, mpbr, poa, poi, point of aim, point of impact, precision, prediction, sample size, statistics, targets


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