When a shot is taken, at any distance, the bullet never goes exactly where we are aiming. There is always some dispersion. Our goal is to minimize that as much as possible, through form and fundamentals or the equipment we are using. As shooters, we need to understand that this dispersion can lead to a probability of hitting or missing targets of different sizes at different distances. In some cases, even if the point of aim were exactly the same for every shot, some would hit and some would miss. Let’s take a look at why.

The Variables That Drive Hit Probability

There are going to be a few things that determine where the bullet eventually impacts and the probability that it will hit the target. What we need to be able to minimize is the error of where the bullet is actually going to hit, and where we think it’s going to hit.

The first one really is dependent on how consistent we are as a shooter. Do we have the fundamentals and control to make sure the rifle fires when and where we want? Any change in our point of aim when the shot breaks will obviously lead to variation at the target. Solid control, a good natural point of aim, and a clean shot break and follow through are all essential, but the best shooter in the world still can’t shoot better than the equipment allows.

The combination of rifle, ammunition, optic, and accessories all has an impact on what kind of accuracy a rifle system is capable of. Advancements in technology and manufacturing have made rifles and ammunition more consistent than ever, but they still aren’t perfect. From a gear standpoint, consistency is the most important thing. We want the rifle and ammunition behaving the same way every time it is fired. Some variations are easier to notice and measure, like bullet velocity measured through a chronograph. Some however, we most likely don’t notice. Things like how well fit together the rifle components are and how the recoil forces move through each component all affect this.

The last real variable is going to be environmental variations. Changes in things like wind speed and direction while the bullet is in flight. This is going to be what we have least control over, and when we look at our hit probability we will assume that the environment stays constant. Of course, if our wind and distance estimates are incorrect then any ballistic data will be incorrect as well and cause issues.

SD and Vertical Dispersion

We have talked about the importance of velocity Standard Deviation (SD) before and this is where it makes a significant difference. When we use a ballistic calculator, velocity is used with the other variables to calculate bullet drop. If we leave all other inputs the same and only change velocity, we will obviously have a different bullet drop.

For an example, try this yourself in the Basic Ballistic Calculator and use just the default values. For a velocity of 2710 FPS we have a drop at 1000 yards of 8.68 MILs (29.84 MOA). An increase in velocity of 10 FPS gives a drop of 8.6 MILs (29.56 MOA) and an increase of 30 FPS (2740 FPS) gives 8.45 MILs (29.07 MOA). These numbers may appear small, .23 MILs and .78 MOA, but that’s a difference of roughly 8 inches at that same 1000 yard target. If you are aiming at the center of a 15″ plate, you have 7.5″ of target above and below your point of aim, and that 30 FPS variation is enough to put you off the target. And that assumes the rifle would have shot exactly where we were aiming.

This is the importance of a small standard deviation. The smaller the SD, the less probability of large velocity changes. Referring back to the SD article, there is roughly a 68% chance that the velocity will be within +/- 1 SD of the average. So if we assume an SD of 10, there is a 68% chance that the shot we take will be between 2700 and 2720 FPS. This gives us a potential drop of 8.75-8.6 MILs (30.08-29.56 MOA) which is around 6″ of potential difference in vertical impact location just on velocity variation alone.

Rifle and Shooter Accuracy

When talking about Velocity SD we assumed the bullet was going exactly to the point of aim, but we know that’s not the case. Most precision shooters consider a rifle that they can reliably shoot 1 MOA or less to be the standard, so we will use that in the example. That 1 MOA (.29 MIL) would be 10.47″ at 1000 yards and with nothing else considered, a 1 MOA rifle should be able to hit a 10″ plate every time if the point of aim is the exact center. Now this obviously means though that any target smaller than that has the chance of a miss just from the group size potential.

Shooters typically refer to what group size their rifle holds, but that only gives us the extreme spread between impacts. Similar to velocity, the extreme spread is helpful to know, but it doesn’t give us any probability of where a bullet will land within that range. There are other measurements such as Mean Radius that give a better picture of bullet impact to point of aim, this will be broken down in a future post. You can also do a SD of bullet impact distance from point of aim that will give its own bell curve and distribution probability. We can then find the probability a shot will be a certain distance from the point of aim.

Combining the Error Sources

Now that we have seen the sources of impact variation we can get from velocity and group size, we need to combine them. We will use the SD of 10 FPS for velocity and assume a truly 1 MOA rifle with a point of impact SD of .167 MOA (rounded). The .167 MOA SD comes from assuming that +/- 3 SD’s will be within that 1 MOA diameter, so 1 MOA/6 = 0.167 MOA.

• 1 velocity SD = 68% chance velocity will be between 2700 and 2720 FPS
• 1 impact SD = 68% chance group size will be under .33 MOA
• 2 velocity SD = 95% chance velocity will be between 2690 and 2730 FPS
• 2 impact SD = 95% chance group size will be under .67 MOA
• 3 velocity SD = 99% chance velocity will be between 2680 and 2740 FPS
• 3 impact SD = 99% chance group size will be under 1.0 MOA

The velocity SD is vertical dispersion from velocity change and we are assuming the group size is centered around the point aim. This gives us a potential impact zone shown in the image below

since we are assuming no wind and perfect conditions, our max vertical variation is a combination of velocity and group size while the horizontal is determined by just the group size. Since the vertical change is a combination of the 2 probabilities, the total standard deviation is given by the equation below.

At 1000 yards, the velocity SD of 10 FPS gives a vertical change of .3 MOA (rounded from .28 above) and the group size SD is .16 MOA. Using the equation, the combined SD is .34 MOA. Note that this is distance dependent since the vertical shift from a 1 SD change in velocity is not the same at all distances. It needs to be recalculated at each target distance.

With our new, combined SD of .34 MOA, we can look at the bell curve probability again. With +/- 1 SD having a 68% probability, we can say that the bullet has a 68% chance of hitting between .34 MOA above or below our point of aim. That would be the same as a 68% chance of hitting a .68 MOA tall target if aiming dead center. This also gives a 99% chance of hitting a 2.04 MOA tall target (3 SD above and 3 SD below point of aim or +/- 1.02 MOA). This is how we figure our hit probability vertically. The same can be figured horizontally if we include the potential error in our wind estimation and how much horizontal shift that could cause. Knowing the wind is between 7-10 MPH will have a horizontal effect on the bullet, but it is different for 7 and 10. The difference between those will be our horizontal variation. The final hit probability is the target size under our new SD bell curve of the combined variables.

Understanding this bullet impact variation can help us answer a few questions. Is a shot worth taking given the odds? If I miss this shot do I need to change something and correct it, or was it purely probability? Knowing what matters, how to control our variables as much as possible, and when we truly need to make changes is how we become better shooters.