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Bench Topicsby Mike Fairbanks

Crunching the Numbers

If you’re anything like me, one of the results of a good day at the range will be lots of data

in the form of a bunch of shot-up targets and a scratch pad full of scribbled notes. With the use of ru-dimentary statistics, some objective sense can be made of these numbers in addition to subjective observa-tions, such as whether a load is clean or dirty or whether it produces a thunderous fireball at the muzzle or seems pleasantly quiet. With-out much calculating, it’s possible to note whether a load is faster or slower than another, or such things as whether it delivers shots into tight clusters or scatters them across the target. However, a gut feeling or simple observation will not give us a hard number to describe the dif-ferences in resultant data. Proper statistical analysis will allow specific comparisons and provide a guide for those seeking top performance from their handloads.

Lucky for us, smart guys devel-oped the branch of mathematics called statistics that specifically deals with (according to Webster) “the collection, analysis, interpretation and presentation of masses of nu-merical data.” While Mark Twain might consider statistics in the same category as lies and damn lies, the use of certain statistical methods al-lows us to make specific conclusions regarding our results. Most of us al-ready do a quick statistical analysis without knowing it when we note

an unusually large extreme spread. Another common and simple op-eration is finding the average value to represent a group of numbers. Unfortunately, other very useful techniques of quantifying the char-acteristics of a set of numbers can get rather involved. Double lucky for us, though, other smart guys have seen fit to include some of these statistical functions in fairly common electronics, such as hand-held calculators, and we don’t have to become math geeks to use these methods.

Before getting too far, however, some basic assumptions regarding our data need to be made in order for our conclusions to be valid. For reloaders, perhaps the most useful branch of statistics deals with data

that is considered to be well behaved or “normally distributed.” Simply, we need to make sure our data com-pares apples to apples. This is actu-ally quite rudimentary and not that difficult to assure. It doesn’t make sense to talk about the average ve-locity of a favorite .45 ACP load and include velocity figures from a .357 while doing the calculations. Well-behaved data is graphically illus-trated by the bell curve, where most of the points tend to cluster around center and yet allow for reasonable outliers. While proof by example does not work in mathematics, enough statistical analysis has been done on firearm performance that we can safely assume typical data will fit within the bell curve.

A basic analysis of normally dis-

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tributed data will involve measure-ments of both central tendency and dispersion. This is a key concept and the reason for using statistics: Sta-tistics measure data. Two common measures of central tendency, aver-age and extreme spread, are quite useful to reloaders. Extreme spread is simple to measure and compare. To find it in a group of numbers, sub-tract the value of the smallest data point from the highest. A large ex-treme spread implies poor precision and can be used to flag a system with significant inconsistencies, whether terrible ambient conditions, bad bul-lets or a loose sight. While extreme spread can be used to single out par-ticularly bad performance, keep in mind that it is based solely upon the two most extreme data points and ignores the rest. This does little to inform us about the nature of a group of numbers other than the ex-tremes. The same phrase, “extreme spread,” is also commonly used to describe groups on target. In this in-stance, a measure of extreme spread is a bit more useful because a load with an extreme spread of 2 inches will always hit a 4-inch bull, so one might look at the measure of ex-treme spread as more of an indicator of whether a load will work or not.

Finding the value of the average for a group of numbers, while more involved than extreme spread, only requires addition and division: di-vide the sum of all the values by the total number of values. This is very useful to reloaders as it provides the single value that every other value tends to cluster around. While the average value does not give an indi-cation of how specific data points are distributed, it does consider each data point. The value of the average can be used to represent each indi-vidual shot, allowing us to quan-tify the number group and make broader assumptions about how fu-ture shots will perform. The aver-age is especially useful in regard to velocity as these readings should be uniform with minimal outliers and

clustered around the average. This velocity average, taken with an aver-age of case web expansion measure-ments, provides us with some of the best indirect indications as to the pressures our loads are developing.

After a quick glance at extreme spreads and calculating averages, measuring the dispersion of data gives us an idea of just how “well behaved” it is. The most common method of doing this is by calcu-lating the standard deviation (SD). This calculation is a bit more in-volved, and I would suggest using a multifunction calculator or com-puter to do the math. This calcu-lation quantifies the amount of deviation from the average of each point of data, a sort of “average de-viation,” if you will. The SD goes hand in hand with the bell curve, where 68 percent (two in three) of the data points will lie within ±1 SD of the average, and roughly 95 per-cent of all the points will lie within ±2 SD of the average. This data forms the shape of the main body of the bell, and the remaining 5 per-cent forms the thin skirt at the base.

That’s all fine and dandy but doesn’t really offer much clarifica-tion, yet. By taking the SD and di-viding it by the average, we come up with a number called the coefficient of variation (CV). This will allow direct comparisons between differ-ent groups of numbers, as the CV is a dimensionless ratio, sometimes ex-pressed as a percentage. I have found that velocity data from good rifle loads will show a CV in the range of 0.10 to 0.80 percent, with pis-tols trending slightly higher. Loads that produce CVs up to 2 or 3 per-

cent can still perform well on target, but anything beyond that gets to be rather inconsistent and starts to show up in poor downrange perfor-mance.

With velocity data, we seldom go beyond extreme spread, aver-age, SD and CV, perhaps due to the fact that velocity is the end result of an incredible number of influenc-ing factors, and we seldom require a very specific velocity anyway. On the other hand, hitting the target means putting the projectile into a very specific spot. The only way to know exactly where a shot will land is to take it, leaving no ammuni-tion for future use. The statistics of target analysis involve mathemati-cal confidence and quantifying such things as reliability and predictabil-ity, and confidence can get rather involved and beyond the scope of this column.

To put it in very simple terms, increasing confidence in future re-sults improves with more samples. The number of shots in a group and number of groups required can be approximated by what kind infor-mation is sought. To make rough sight adjustments or to get an idea of where groups will typically im-pact, a few groups will suffice. Again, this is taking more of a “does it work or not” measure like extreme spread. Quantifying the influence of a minor component change, such as comparisons between primers, will require many shots into many groups. When comparing factors that have a minute influence on a load, a subjective conclusion uti-lizing the brain’s amazing ability to process multiple inputs without

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knowing exactly how or why is as good a method as any.

Now let’s imagine we just got home from the range, have cleaned up and are ready to take a closer look at the raw data. For this ex-ercise, here are the actual numbers from two different loads fired in the same SP101 .357 Magnum. For the sake of comparison, the only differ-ence is the powder charge, a factor that can have a substantial influence on performance. The rest of the load consists of an 18 Brinell hard-ness number (BHN), 125-grain cast Lee flatpoint, sized .358 inch, lubed with Lyman Moly, loaded to 1.560 inches OAL in Federal brass and

sparked with Federal 100 Small Pis-tol primers.

Right away, we can guess that the Universal load was much more consis-tent – but by how much? And, what is the average velocity of each load?

Extreme Spreads:Load 1: high of 1,037 - low of

806 = 231; not that goodLoad 2: high of 1,172 - low of

1,142 = 30; much betterAverage Velocities:Load 1: 947.7 fpsLoad 2: 1,157.8 fpsStandard deviations (courtesy of

my calculator):Load 1: 68.6 fps; 95 percent of

all shots should be between 810.5 and 1,084.9 fps (±2 SD)

Load 2: 7.8 fps; 95 percent of all shots should be between 1,142.2 and 1,173.4 fps (±2 SD)

Making a direct comparison of the standard deviations does not take into account the differences in average velocity, so if we create a dimensionless ratio by dividing the SD by the average, we get the CV:

Load 1: 68.6 ÷ 947.7 = 7.24Load 2: 7.8 ÷ 1,157.8 = 0.67Now, a direct comparison can be

made. We see that the CV of load 2 is much less than load 1. At this point, a dyed-in-the-wool statisti-cian would do the analysis required to measure just how much more consistent load 2 is, but for our pur-poses, it is simply enough to note that load 2 appears to be a much better performer than load 1. The differences in consistency in veloc-ity also showed up in the groups on target. Load 1 did well to print into 6 inches at 25 yards while load 2 cut nice 1½- to 2-inch clusters. For whatever the reason, I think load 1 just didn’t develop enough pressure to make Unique burn consistently; it needs help. Perhaps a stronger primer or tighter crimp would help. By using statistical analysis, we’re able to quantify what we might al-ready recognize, that load 2 clearly performs much more consistently. Now, instead of entering all the raw data into my data book, the result of this load comparison is recorded like shown in the table. A quick glance at the CV and accuracy columns tells me the Unique load is not a top performer.

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