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 in yellow. 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

Hi PAUL,

Thank,s got the precision and accuracy,makes sense to me,you dont tend to look at point of aim when testing new rounds just how close together they are.
I am OK with imperial measurements in fact being from the premetric generation I can visualise feet and inches better.
The graphs are making more sense now that ive seen the two groups overlayed around that common PoI
Looking forward to the next instalment,thanks again............................Richie

OK... Lets get down to the math and stats.
 
Oh yeah, I forgot to mention my favorite quote:
"There are liars, damn liars, and statisticians." Mark Twain.
 
Words to keep in mind at all times when someone is trying to 'sell' you numbers. You should treat me and my numbers with no less skepticism.
 
Below are the sets of calculations for each of the above computer-generated graphs (targets). Let's go over each equation and what it means and why I include it in my analysis.
 
The THL editor may allow me to write equations in mathematical format here in the text. If so good, if not, I'll do my best to represent them so that there's no confusion about which one I'm referring to.
 
First on the page comes the greek "tau" (The editor won't express it in its greek form - it will look like "t".) Tau is called "student's "t". It has statistical historical importance that I can explain if anyon'es interested, but for now, it is a variable that describes certain characteristics about a Gaussian distribution based on sample size. Note that it is different (4.303) for the combined graph whose sample size is "6" instead of (2.571) for the first tow graphs with sample sizes of "3". This is why (mathematically) sample size is VERY important.
 
 
The variables that look like "u" sub "x" and "u" sub "y" are actually "mu sub x" and "mu sub y". "Mu" is the greek character most often used in statistics to denote "mean", another word for average. So... "Mu sub x" and "mu sub y" in this document are used to define the average value in the x and y axes, of the position of the bullet holes. Mu sub x, Mu sub y defines the mathematical center of the group.
 
 
For group #1:
"mu sub x" = 0.833"
"mu sub y" = 0.626"
 
Meaning that the center of the first group was 0.833" high and 0.626" right of the PoA.
 
 
"Maxspread", is something I've just very recently added to suit those here at THL that would like to have "max spread" values. It's really not a useful stat for me. It is simply calculated by taking the square root of the absolute value of the differences between the max and min x-values, plus the absolute value of the difference of the max and min y-values. This is sometimes called the "euclidean distance", as it uses the Pythagorean thoerem (the square of the hypotenuse of a right triangle is equal to the sum of the squares of the height and base), to find the value. It has NO predictive value - protestations by "gunwriters" and benchresters notwithstanding. This will be clear (hopefully) soon.
 
 
"Maxspread" for this group is 1.142".
 
 
 
"Sigma sub x" and "sigma sub y" are the standard deviations of the coordinates of the shots in the x and y axes. This statistic, coupled with the average (mean)
provide all the information we need to predict (estimate) were the next one or one thousdand bullets will hit.
 
 
"Sigma sub x" = 0.345"
"Sigma sub y" = 0.484"
 
"Meaning" that the distribution of the bullet holes is about 1.5 times as high as it is wide. These variables have greater value later.
 
 
The "delta" variables, "delta sub c", "delta sub e" and "delta sub ea" are, in order of appearance:
 
The average distance from the PoI the shots were.
 
The average euclidean distance from the PoA the shots were.
 
And the euclidean distance of the average distance from the PoA the shots were.
 
Mostly, they're just mathematical 'fiddling around'.
 
 
"xaxis" and "yaxis" are the names of the variables that contain the lengths of the x and y axies of the 95% prediction ellipse. Here's where 'things' get useful. They are arrived at by taking "tau" times the standard deviation and adding and subtracting that value from the average (the center of the group). Remember that "tau" is a function of sample size. The smaller the sample size is, the larger "tau" is. The larger "tau" is, the larger the length of the axis is going to be - meaning the more uncertain you are about the group actually representing "all possible" groups.
 
"xaxis" = 2.966"
"yaxis" = 4.196"
 
Meaning that when we draw the ellipse that decribes the 95% probability of the next shot, these values will be the lengths of its axes.
 
 
"Area" is the area in square inches inscribed by the 95% confidence ellipse. This is the number I'm interested in. This number tells me that if I shoot 100 more bullets, the average PoI will fall within an ellipse with that area. If I change "tau" by reducing the sample size by one, I get the 95% Prediction ellipse. This ellipse is the area in which I can expect 95 of the next 100 shots to land.
 
 
So, let me illustrate the importance of sample size.
 
In group #1, the sample size is 3, "tau" is 4.303, and Area is 9.713 square inches.
 
 
In group #2, the sample size is also 3, "tau" is also 4.303, but Area is only 4.856 square inches. The reason group #2's Area is smaller, is because group #2's standard deviations are smaller. Group #1's are:
x - 0.345"
y - 0.484"
 
 
Group #2's are:
x - 0.267"
y - 0.313".
 
(Of course you can "see" that group #2 is smaller than group #1, but these numbers allow you to quantitatively compare how much larger group one is than group 2.)
 
 
However, look what happens when we include all the shots and compensate for 'scope adjustments; sample size is 6, "tau" lowers to 2.571, and Area drops to 2.096 square inches, even though the "maxspread" is even LARGER than it was in the largest 3-shot group.
 
 
Let me put this is practical perspective. Let's say you and I were at the range, and I just watched you shoot the second group. At that point I ask you if you wanna bet on where the next shot will land. You say "Sure. Since the max spread of the group is only .821", Ill bet the next shot will fall within half an inch of where I aim." (We're assuming here that your rifle is sighted in to shoot at Point of Aim.) I'll take that bet over and over, because for every 100 of those bets I make, I should win 95 of them. Now, that said, every time you take another shot, I'll recalculate the 95% prediciton ellipse, and bet accordingly.
 
 
Betting and bragging aside, what's truly of value to me, is 1) having the best guess possible on where the next shots out of this rifle are going to land, and 2) being able to objectively (as opposed to emotional, or "wishfull thinking" guesses) compare loads and bullets for this rifle.
 
 
That's one piece of three that I use to evaluate my firearms. Next comes "The Figure of Merit".
 
 
(You may not think so, but this is really the Reader's Digest condensed version".)
 
 
Paul

PS - I don't know why the THL editor doesn't like the order I want images. At any rate, group #1 is first, the combined shots are second, and group #2 is last.

Richie, I didn't see your post between my first and second ones.
 
So... on to Figure of Merit.
 
Now that we have an objective and accurate measure of the precision of our rifle, load, and self, we can, with some confidence, predict where the next bullet will strike when shot from the same distance. However, not all 'things' are going to be shot at the same distance, and more importantly, bullets behave differently when shot at different velocities or impact at different velocities. Enter my "Figure of Merit"calculations.
 
A brief (very brief) history. Back in the late '70's , nobody had personal comuters, let alone ballistic calculators. However, I was a graduate student and had access to computers, and was taking high-level math and physics. I wanted to "do" personal ballistics calculations. All my math and physics profs glossed over the 'gory details' of external ballistics, using a common phrase - "We'll assume there's no air resistance." Yeah well... that's a BIG assumption. It does allow discussion of concepts and simplifies the math, but it departs from all semblence to reality. Also, I was finding that my personal data from range sessions rarely fit the results predicted by the ballistic tables found in the backs of reloading manuals. So... I wrote my own external ballistics program. Doing so involved a great deal of research into the mathematical and practical history of ballistics. It was tedious and frustrating, but ultimately gave me a very strong foundation in external ballistics. Furthermore, it gave me results that coincided far better with my personal range results.
 
Today however, there are many external ballistics programs available, and just about all of them are far more user-friendly and have lots more options available than mine. However, I'm primarily interested in calculating trajectory, retained velocity, retained energy, and momentum, and my original program - which you will see here - does that just fine. More importantly, it calculates my Figure of Merit.
 
Again, from the above analysis of target data, we can obtain the area of the 95% prediction ellipse. Using that as an objective measure of precision, and adding muzzle velocity data, I can predict (estimate) where the bullet is going to hit at any reasonable range. If I put in other information, I can start to build a fairly complete picture of the performance of me with this rifle.
 
I'm going to continue to use Sav17's info. Where there are data values missing, I'll make assumptions. He can clarify.
 
Below you will find two Figure of Merit tables. The first is the Maximum Point Blank Range Table. I assumed a muzzle velocity of 2900 f/s (derived from Hornady load manual), a gun weight of 8.5 lbs, a range temperature of 50 degrees F and a range altitude of 100 feet. The MPBR for a 6" target is 266 yds. This means that this bullet sighted in at 228 yds will be 3" high at 133 yds and 3" low at 266 yds. At 266 yds, the bullet will have a velocity of 2256 f/s, a momentum of 1.002 slug-ft/s, and a ke of 1130 f-lbs. There will be 8.2 lbs of "free recoil", the bullet will cross the line of sight at 25 yds and 228 yds. The Figure of Merit is 866. I'll explain later.
 
The second FoM table is for a fixed range of 300 yards. Velocity at 300 is 2181 f/s, momentum at 300 is 0.968 slug-ft/s, KE at 300 is 1056 ft-lbs. Most of the other figures remain the same. Note that the Figure of Merit is now 784.
 
I'll explain the FoM in detail later, but in the mean time, the figure represents how the gun in question compares to a .30-06 using a 180 grain bullet with a muzzle velocity of 2700. The 30-06's  arbitrarily chosen FoM is 1000. More will be clear later. Look over the pictures and ask away.
 
Paul

Still appears as though I would not enjoy being 300yds down range from the puny .243 with a 100gr bullet
Don;D

Hi Paul,
the mist is beginning to clear,I need to print this off to study it properly,can you explain
M.P.B.R to me I just cant seem to grasp it.
By the way thanks for the time and effort you are putting here I hope this is not an inconvenience for you
Your friend..............Richie

Maximum point blank range is the maximum range at which one can aim "dead on" and not miss (due ot the arc of the trajectory), a target of pre-detemined size.
 
What that means is; once you pick a target size, which, in the case of my FoM calculations, is 6" (except for the .17 Remington which is 4"), you adjust the "zero range" so that the bullet is no more than 3" high. Once that zero range is determined, you look at the trajectory table and see where the bullet is 3" below the line of sight. That range is the MPBR.
 
Theoretically, out to that range you can aim "right in the middle" of the critter, and as long as its "kill zone" (chest cavity) is larger than 6" vertically, you will hit "in the kill zone". Not too much value to the number, except for comparison's sake. I'm more inclined to set a personal maximum range based on my ability to hold the rifle steady, and adjust my aim for the actual range. MPBR is useful in conversations that have "My gun's better than your gun", (in whatever form it might take), in them.
 
Paul

Richie;
There is a fairly long thread on mpbr a few months ago, basicly, just decide how high the vital area in the game you are after, say 8" for the heart/ lung area of an average size deer, take 1/2 that, 4", sight the rifle so it will hit no more than 4" high and no more than 4" low, a ballistics table will show this easily, Pointblank is a site where you can get a free download of a MPBR calculator.
Look at Paul's graph, starts out 3" low, crosses the line of sight, increases elevation to 3" above line of sight, decreases to cross line of sight again and finally runs out at 3" below line of sight.
Don

thanks guys,

got it now,this explains the advice I was given to zero my 223 1" above centre at 100 yards for the 223
...........................................Richie

Thanks guys,

the mist is definately beginning to clear,I have downloaded the pointblank software and studied gitano,s math,also got the rcbs software and things are now becoming clear.
Theres more to it than pointing the rifle pulling the trigger and waiting for the bang.
Education is a wonderful thing;) :D :D
Cheers guys, your friend..........................Richie

Hi Guys,

I now realise where my confusion was created,I wrongly thought that the bullet travelled in a straight line from below the scope to converge with the point of aim.So in my tiny confused mind the bullet travelled from below the scope on an upward trajectory until it converged with the point of aim and continued on this upward trajectory until it lost momentum and started to drop(please dont laugh duhh!!!)in effect ,if the rifle was zeroed at 100 yards you would be low to 100 and high afterwards,oh dear i feel so thick:o .
But now you have shown me that the projectile follows an arc "eureka" it all becomes clear,just goes to show really without the very basics you will never be able to understand,like trying to read before you learn your ABC,s. :o :o Your enlightened and much more knowledgable friend..................Richie