Marc’s article talking about chronographs got me asking the question. How many shooters using chronographs understand the readouts?
Some of the readouts are self-explanatory. The first readout you will see the measured velocity of that shot. It is a single reading, and it reflects the shot it just measured. The next reading usually is an Average. This is also pretty self-explanatory; I feel like most shooters who own a chronograph can grasp the concept of Averages.
A brief word on Extreme Spread
However, the last few numbers may give a few shooters some pause. Usually this is extreme spread, and standard deviation. We’ll start with extreme spread as it is the easiest to look at.
Extreme spread is the maximum recorded velocity minus the minimum recorded velocity. This number only updates when there is a new highest velocity or a new lowest record velocity. So it’s not uncommon if the number does not update very often.
If you were to picture extreme spread on a target, it would look like this:
The blue circle represents the average velocity. It is closest to the point of aim, which is the center of the target. The max velocity shot is the red dot at the 12 o’clock position. It shoots high compared to the group average. The lowest velocity shot is the 6 o’clock position and it shoots the lowest on the target compared to group average. To put it succinctly the extreme spread you will likely witness on the target, as it is derived from measured values and not from a statistical model.
However Standard Deviation is harder to visualize as it is not something you can see easily on the target. It is typically either a two digit or single digit number that the chronograph displays. Most shooters know that the lower the number the better, but they may not know what the actual real-world effect is or how they can use it to qualify the quality of their ammunition.
Setting up the Data
For the examples below I grabbed some of my 338 Lapua Data. This was loaded with 300-grain Sierra MatchKings and fired from a 24-inch barrel. The data set is for 20 shots, which is probably more than what most people use. Typically, you want to have at least 10 shots before you consider the Standard Deviation measurement reliable. The larger the sample size the more reflective the Standard Deviation measurement will be.
How and what I am using to generate the data is not important. You can apply all of the same principles on a 32 ACP shot from a Beretta Tomcat.
So what is Standard Deviation?
It is a measurement used in statistics to communicate variation within a data set. The lower the number, the lower the variation within the data set. In our case, the lower the number the more consistent the ammo is, and if consistent velocity is what makes a tighter group, then a lower velocity standard deviation will produce a tighter group.
For our purposes when we use standard deviation, we can treat it as a tolerance. So, if the standard deviation comes back as 11 fps, then we can read that as ±11fps. That means that if our average velocity is 2613fps, then our tolerance on that measurement is ±11fps. We can reasonably expect that the majority (67%) of our shots to land between 2,624fps at the high, and 2,602fps at the low.
Notice how I said 67%, that means our ±11fps tolerance accounts for about 7 out of 10 shots. We can reasonably expect that 3 shots will be faster than 2,624fps or slower than 2,602. So how do we account for those shots? Simple, we double the standard deviation number. In our case that is easy, ±11fps becomes ±22fps. Our new max velocity is 2635fps and our new min velocity is 2,591fps.
When we double the standard deviation, we can now account for 95% of our shots. That means that out of 100 shots we can reasonably expect that 95 of them will fall between 2,635fps at the high, and 2,591fps at the low. That is pretty good, most precision shooting competitions are less than 100 shots, so it’s likely you’ll only see a few errant fliers.
What if you wanted even more guarantee that you aren’t going to see any fliers due to unpredicted changes in velocity? Well, we can add another level of standard deviation. Instead of ±22fps, we add an additional 11fps for a total tolerance range of ±33fps. Our new max velocity would become 2,646fps and 2,580fps. This covers 99.7% of our shots fired. In 1000 shots we would only expect to see 3 shots fall outside this window.
So what does that look like on Target?
The danger of statistics is we can see all these numbers and believe they actually mean something. The reality often is dependent on the answers of two questions.
- Is it statistically significant?
- Is it practically significant?
To illustrate this, I drew up a target in excel and I plotted the estimate impact point at 1,000 yards. Why 1,000 yards? Because even at this distance the differences of an 11fps standard deviation can be hard to see on the target. At distances closer than 800 yards we can look at the 11fps standard deviation and know that velocity variation is there, it’s more than 0 so it’s a statistically significant number, but the practical effect on target is minimal so we can ignore it.
This is true for this load, but if I was shooting a bullet shaped like a soup can, the differences in velocity might have a much greater effect. On the target below there are three concentric rings. The center ring is 1 Minute of Angle (MOA). At 1,000 yards this circle is 10.47in in diameter. The middle circle is 2 MOA and the outer circle is 30 MOA.
To illustrate how standard deviation affects the load we plotted out a series of dots which represent the predicted bullet impact if velocity variation was the only thing impacting trajectory. This is called a single factor analysis, we all know there are lots of other covariables that determine the actual shot placement. This does not account for them. It is useful only to show how group sizes may be impacted by variation in muzzle velocity.
The blue dot in the center is where we would expect the average velocity bullet to impact, provided the shooter dialed up his scope 34.25 MOA. You can see it’s slightly above the center point on the target, that is because at this distance ¼ MOA “clicks” are starting to become a little bit too coarse.
The two green dots are the extreme spread we would expect to see with 1 standard deviation or a velocity window of ±11fps. This gives us a max velocity of 2,624fps, and a min velocity of 2,601fps. We would expect about 67% of our shots to fall between these two dots, with all other things being equal.
The pair of yellow dots represent 2 standard deviations. A velocity window of ±22 fps accounting for 95% of the shots fired. The min velocity for this window is 2,590fps and the max is 2,635fps.
Lastly, we have our pair of red dots which represent the 3 standard deviations from the average. This results in a velocity window of ±33fps, with a max velocity of 2,645fps and a min velocity of 2,579fps. This accounts for 99.7% of shots fired, or to put another way out of 1000 shots only 3 would be expected to count as “fliers”.
How do we use standard deviation to determine good enough?
You will hear shooters bragging about having standard deviations in the low single digits. That is either an extremely consistent load, or they only shot about three shots over the chronograph. Regardless, when is the number low enough?
In the above example we had a standard deviation of 11fps. This equated to a 6.58in circle at 1,000 yards which means that 67% of my shots will form a sub-MOA group if everything is perfect. (We know it never is, but set that aside for the moment). 95% of my shots, which include the previous 67% will land within a 13.5in circle which is about 1.3 MOA and if I really want perfection then I must move to 3 standard deviations which means that 997 out of a thousand shots will group within a 2 MOA circle.
Obviously, you must figure this out for your load and what distance you plan on shooting at with what rifle. A simple ballistics calculator can be a decent tool to play with. Change only the velocity of the bullet for the velocity with the standard deviation applied. Then look to see how that shifts the impact point down range. Recording the highest and the lowest values may provide you a reference as to what you might expect the minimum groups size to be.
Again, it’s not perfect, but it’s a starting point.
A Tale From the Real World
In early 2000’s Special Operations Command (SOCOM) put out a request for a new precision rifle system. Several companies produced rifles for the trial including Remington Defense which was a subsidiary of Remington Outdoor Company. The contract included both the rifle and the ammunition to form a weapon system.
This weapon system had an accuracy requirement of 1 MOA at 300, 600, 900, 1,200 and 1,500 meters. This measurement was done by firing 150, ten shot groups. No group could exceed 1.5 MOA of extreme spread. All of the accuracy testing was done at 1,500 meters (1, 640yds). The total course of fire at 1,500 rounds. Remember back when we did our basic 3 standard deviation that we could expect 3 rounds out of 1000? This was a $80 Million dollar contract that relied on the performance of $11,000 of ammunition.
On review the engineers working on the ammunition side of the project did a simple calculation. Very similar to what we just did. They came away with numbers that showed that velocity variation in the ammunition alone, was enough to eat up roughly 75% of that window at a single standard deviation. This meant that 495 shots had the potential of opening up the group size to where it would fail the test.
This left only 25% of variation provided to the weapon system itself. This could be heating and cooling of the barrel, differences in how the bolt closed and positioned the round in the chamber, etc. There was a legitimate argument on whether to use 1 standard deviation, 2 standard deviation or 3 standard deviations in order to say the ammunition met spec. Remember millions of dollars were on the line, so 3 shots out of 1000 that you and I would probably be fine living with, were real risks to the company.
For reference, ±11 feet per second standard deviation I have on my handloads, is pretty representative of what some of the best quality factory loaded ammo will achieve. What the company needed to produce was ammunition with a standard deviation that is half to one third of that in order to have a high probability that velocity variation alone would not cause rounds to fall outside the contract spec. This is achievable, with very finely tuned handloaded ammunition, it is not a scalable process.
Remington Defense did a lot of work to try and figure this one out including cooking up some very consistent and accurate hand loads to try and meet the conditions of the contract. Yes, it was shady, as hell. Initially they won the contract, beating out Barrett and others. However, in 2018 it was decided that the weapons system did not conform to SOCOM requirements and Remington Defense lost the contract. Due to this and other unfortunate missteps the Remington Outdoor Company filed for bankruptcy a few years later.
Standard Deviation Wrap Up
Standard Deviation is not typically going to be really useful for you when developing plinking loads or trying to shoot some groups with your pistol. However, if you are shooting for distance and you want to understand how velocity variation will impact your group size, standard deviation is your best friend. We have tried to explain it in very simple terms without going into all of the math, and graphs that usually comes with explaining stats. Hopefully in a way that makes sense to the “average Joe”. If you have questions, let me know.
-Jay-