I have been working on an article dealing with statistical analysis of ammunition velocity and accuracy. You will soon be able to access it from the articles page. Here is a preview that deals with velocity.
When I encounter a new factory load or prepare a new handload, the first thing I want to know is the load’s velocity. Sometimes I am making a comparison and I want to know whether load A is faster than load B. Modern electronics have made accurate velocity measurement fairly easy to accomplish. All you need is a good chronograph and a quiet day, and, of course, your rifle.
Scatter
Open a new box of factory ammo and you will see a shiny bunch of cartridges that look identical. Fire them in your favorite pole, however, and you will find a range of velocities. In other words, there is “scatter” in the shooting results. A statistical discussion may use this term or may use the more formal term “dispersion.”
How, then, do we decide what the velocity of a load really is? We are aided by the fact that the scattered values usually have a tendency to cluster around a certain value in the range. Statistical treatment calls this the “central tendency” and if we can identify it, we will have something to hang our hat on. The most common measure of the central tendency for shooters seems to be the “arithmetic mean,” and it is just fine to call it the “average velocity.” To get it, sum the values and divide by the number of shots. Of course, your chronograph will happily give you this value after you have shot your string.
A typical example of the treatment is shown by velocity data for a handload that I examined recently, 35.2 gr of Hodgdon LVR powder pushing a Hornady 160-gr FTX bullet. Eight rounds fired from a Remington Model 788 gave the following velocity values:
2372, 2356, 2346, 2345, 2314, 2229, 2217, 2180 fps
The mean (average) velocity is 2295 fps. This data set seems to be flawed because it has a large spread of 192 fps, the result of a couple of abnormally low values, called “outliers.” Outliers are encountered frequently in a set of measured velocities. I might be tempted to throw them out, and calculate the average for the remaining cluster, but that would not be playing fair and cannot be allowed. These are the measured values and I can only assume at this point that they reflect the true performance of the load. The range is what it is and it does give me some preliminary idea of the uniformity of the load.
Now I would like to examine an alternative kind of central tendency, (average) that is called the “median value.” To get the median, list all of the velocities in order and find the value that has an equal number of velocities above and below it. That is the median value. If the list has an even number of velocities, the median is the average of the two middle values. The median value for my handload is 2330 fps, a value considerably higher than the mean value, 2295 fps. The median is seen to minimize the effect of the outliers.
As noted above, one or more outliers will often be observed in a set of measured velocities. It is my opinion, then, that the median value is a better measure of a load’s tendency than the mean value. However, the mean is the value more often used for a velocity average in the shooting literature. Sometimes the difference in the two measures is rather small, but sometimes it is not.
Degree of Scatter
The mean or median values tell us little about the amount of scatter in our velocity data. Of course, we would like our cartridges to give us a uniform, that is, a tightly clustered, sample set of velocities
One measure of the amount of scatter in a group of shots is called the Standard Deviation. A regular part of statistical treatment, the SD is calculated using squared values for the velocity differences found when comparing the individual shots with the mean value. The Standard Deviation is the scatter measure most used for shooting results because most chronographs calculate SD for you after you fire a string of velocities. Thus, you can get an SD value without knowing how it is derived or what it means. The standard deviation for my handload above is 74 fps
A drawback of the SD value is that its meaning is not very intuitive. It does not give an easy mental picture of its meaning. A strong point is that it can be used for additional calculations to determine whether the means of two series are really different. This is especially useful when comparing the velocities of two different loads.
An alternative, more concrete measure for scatter is the “Average Deviation.” To get it, determine the difference of each velocity from the mean velocity. For the handload above (mean velocity 2295 fps, median velocity 2330 fps) the list is:
2372, 2356, 2346, 2345, 2314, 2229, 2217, 2180 fps
Deviations from Mean: +77, +61, +51, +50, +19, -66, -78, -115
Deviations from Median: +42, +25, +16, +15, -16, – 101, -113, 150
Considering all differences to be positive values, calculate the average. The smaller the average deviation, the more uniform the load. The average deviation may be calculated for the mean or for the median. For my .handload data, the values show 65 fps average deviation from the mean, and 60 fps average deviation from the median. The rather clear picture given by the AD is that an average shot with this load will fall about 60 fps from 2330 fps.
What If I Am Dissatisfied?
Although I think I have made the best analysis of the data I have to work with, I am not very satisfied with the outcome. Do I have recourse? Yes, fire more shots. The reliability of the mean or median will always be improved by additional data points. This will clarify the tendency toward outliers and improve confidence in the average. Don’t have the time for more shooting? More shots too expensive? OK, then try the analysis with median and average deviation outlined above.
Working with medians and average deviations involves a bit more work, but it can be worth it.