by Ken Howell

Y**OU’RE NOT** going to believe how much more your chronograph can tell you about your handloads. And you’re going to think that I’m a very clever old coot to have discovered these nice little gems that I’m going to pass along to you now. Hold that lovely thought ?I’ll set the record straight on it, in due time.

*Secret 1.* With only five test rounds ?one round each with five different powder charges ?and a little simple math, you can find-out as much about your load’s velocities as you’d learn if you’d fired five ten-round strings.

*Secret 2.* With ten test rounds (all alike this time) and some more simple math, you can find the true velocity of your load and how widely the “off” velocities vary from the “true” velocity ?as if you’d fired a hundred or a thousand test rounds.

*Secret 3.* With ten test rounds and a little more math ?a bit more complicated this time ?you can convert your chronograph data into surprisingly close estimates of your chamber pressures. This technique will have to wait for another article, but I want you to know now that it’s in the offing. Now let’s get into the details of a couple of these neat little secrets.

A loading manual, let’s say, tells me that 58.0 grains of a certain powder is the recommended maximum in my cartridge. The velocity that the manual lists for this load is essentially meaningless to me. The same load in my rifle may give me a good bit more or a good bit less velocity than some experimenter got with that published load in whatever rifle that he was shooting. I want to know what I can expect that load to do in my rifle.

So I load just five test rounds ?one round with 50.0 grains, one round with 52.0 grains, one round with 54.0 grains, and one round each with 56.0 and 58.0 grains. (If the maximum load were only thirty grains, I’d load my test rounds in one-grain increments. In even smaller cartridges, I’d go down to half-grain or even tenth-grain increments. If the maximum load were eighty or ninety grains, I could load my test rounds in three-grain increments.)

Then I fire these five test rounds over the chronograph screens, record the velocity of each round, and figure the first, second, third, and fourth differences between these velocities and their differences, like this (paying close attention to the positive [+] and negative [-] differences):

charge(grains) |
velocity(ft/sec) |
first difference(ft/sec) |
second difference(ft/sec) |
third difference(ft/sec) |
fourth difference(ft/sec) |

50 | 2,415 | ||||

+111 | |||||

52 | 2,526 | +12 | |||

+123 | -17 | ||||

54 | 2,649 | -5 | -26 | ||

+118 | -43 | ||||

56 | 2,767 | -48 | |||

+70 | |||||

58 | 2,837 |

In the table above, the difference between 2,415 ft/sec and 2,526 ft/sec is +111 ft/sec, and the difference between 2,526 ft/sec and 2,649 ft/sec is +123 ft/sec, and so on. (With me so far?)

The difference between these first two differences ?+111 ft/sec and +123 ft/sec ?is +12 ft/sec. Carry this same scheme on out to the right, and the difference between +12 ft/sec and -5 ft/sec is -17 ft/sec. And on we go: the difference between -17 ft/sec and -43 ft/sec is -26 ft/sec (the fourth difference, which I’m going to use in a minute).

As I fire these five test rounds, of course, I check my fired cases just to make sure that they show no sign of a pressure problem. When I’ve fired all five of my test rounds, I’ve worked up to the recommended maximum charge without expending more than one primer, one powder charge, and one bullet at each level.

Now ?what can I learn, that’s truly useful to me, from these five test rounds?

*I can now calculate an accurate velocity figure for the middle charge ?54 grains ?as accurately as if I’d fired five rounds with 54 grains.*

How?

From the velocity that my Oehler 35P records for the middle charge (2,649 ft/sec), I subtract 1/12 of the fourth difference ?1/12 of -26 ft/sec equals -2.2 ft/sec, and subtracting this negative figure is the same as adding it as a positive figure, so 2,649 ft/sec plus 2.2 ft/sec equals 2,651.2 ft/sec.

I can go further and figure the standard deviation (23.4 ft/sec). Still further: 2,651 ft/sec divided by 54 grains tells me that I’m getting a return of 49.1 ft/sec for each grain of powder that I’ve invested in that 54-grain charge. So let’s say that for some reason I want my velocities to average an even 2,700 ft/sec. I divide 2,700 ft/sec by 49.1 and learn that 54.98 grains (55.0 grains on the powder scale ?three grains below the book maximum) will give my bullets the 2,700 ft/sec velocity that I want.

Mathematicians will recognize this as an application of the familiar (to them, not to me!) “method of least squares.” (I’ve seen the term used ?’nough said.) The rest of us can use it whether or not we know what to call it or how it works. We know that we get variations ?from one round to the next ?that we can’t reduce to rigid uniformity or consistency, but with this simple routine, we can use those variations to tell us something that’s useful for us to know.

I’m not smart enough to know whether this notion is sound or stupid, but I suspect that the “least squares” method can also be helpful to improve the usefulness of the data that you get from a chronograph that isn’t as good as an Oehler.

*Second secret:* now here’s a puzzler. I’m going to play a little trick on you for a minute or two (forgive me, please) to illustrate a point and to make it easier for you to see it and to remember it.

I sit down at my little shooting bench and fire one round over the chronograph screens ?2,427 ft/sec, my Oehler 35P tells me.

I get up, and my pal Mick sits down and ?with the same rifle ?fires one round of the same ammunition. The Oehler says that the velocity of that bullet was 2,439 ft/sec. *Hmmm.*

Then Al takes a turn ?one round ?2,398 ft/sec.

Bill fires a round that records 2,445 ft/sec, and Lew’s round goes 2,404 ft/sec.

What the heck’s happening here? Our velocities average 2,422.6 ft/sec, but none of our rounds has recorded that calculated average of 2,422 ft/sec. We’ve been firing rounds from one batch of cartridges, all of which one experienced handloader carefully loaded, to be as nearly identical as possible. Our standard deviation, the Oehler’s print-out says, is 18.66 ft/sec.

Let’s try this again. In the second go-’round, I get 2,410 ft/sec; Mick gets 2,398 ft/sec; Al gets 2,421 ft/sec; Bill gets 2,381 ft/sec; and Lew gets 2,415 ft/sec.

Our average is now 2,405 ft/sec, and our standard deviation this time is 14.18 ft/sec. And not one of us has fired a round at the calculated average velocity of 2,405 ft/sec.

Have you figured this one out? This trick “problem” is only a typical, normal test situation. The trick that I played on you was saying that a different shooter fired each round in each of two five-round strings. In the actual test that I’m talking about, *one* shooter fired *all ten rounds* in a *single* ten-round string. I’ve just averaged the first and last five rounds separately. The real shooter’s ten-round average was 2,413.9 ft/sec, and his standard deviation was 18.77 ft/sec.

However carefully we handload our ammo, we have no way to load cartridges that will give us absolutely uniform performance from one round to the next, in either velocities or pressures. And if our cartridges happened to be perfectly uniform, our very finest (but not perfect) recording instruments would record them as slightly different. When we fire test lots, we’d like to know whether one test lot is more nearly uniform than another. But with all these variations, how can we tell?

Firing larger and larger test lots would bring the average of the total number of rounds fired closer to what we might call our “true velocity” ?that theoretical velocity, which may not exist for any single round, that splits all the variations right down the middle, so to speak. But there’s no need to fire mountains of test rounds. Math lets us figure our true velocity as accurately as if we’d fired at least one teeny mountain of test rounds.

If we compare each actual velocity to the calculated average velocity, we get the deviation of each shot from the average velocity. If we add all these deviations and divide by the number of rounds that we’ve fired, we get the mean deviation. But this isn’t a lot of real help to us. So we multiply each deviation by itself ?12 ft/sec times 12 ft/sec, for example ?add the products of these squared deviations, divide by the number of rounds that we’ve fired, and get a number that’s called the variance of these deviations. The square root of this variance ?just punch the right button on a good pocket calculator, and you have it ?is what the statisticians call the *root main* (or is it *mean*?) *square deviation* ?fancy term, huh? The favored term for it, these days, is the *standard deviation*, and it’s generally regarded as the best way to compare results that vary mathematically ?group sizes, velocity spreads ?to compare relative consistencies.

And I could go on ?if I were a mathematician and knew how to divide the standard deviations, square the result, refer to a table of variance ratios, and apply something called the *F* test ?or if I knew how to convert the standard deviations to estimated standard deviations, to multiply the difference between the averages by the number of rounds fired (easy, I understand, with the appropriate tables), and to apply the *t* test to find ?finally ?for example, that a difference of, say, 17.6 ft/sec is only routine and to be expected in five-round samples of a given load.

Another example: in 1964, a shooter’s five-round average velocity was 2,826 ft/sec. In 1965, after firing between 1,000 and 1,500 rounds of the same ammo, he fired another five-round test string ?at a significantly different air temperature ?and his average velocity was then only 2,716 ft/sec ?a “loss” of 110 ft/sec? Not really. He did all the right figuring, applied the *F* test, and discovered “with a high degree of confidence” that this difference was only the normal, typical, to-be-expected result of a five-round test of the uniformity of his load. Also, the *t* test revealed that there was less than one chance in a thousand that the “loss” of 110 ft/sec was only a normal variation in his load. The rounds that he’d loaded were as nearly identical as anyone could have made them.

The lesson that we math ignoramuses can learn from all this is that *different numbers in our chronograph data don’t necessarily mean significantly different performance that we should try to refine or expect to reduce further.* The ultimate in practicably reachable uniformity or consistency is always going to give us a bunch of numbers that vary “all over the place” while they’re essentially identical.

I’ve borrowed the thunder of another man’s brain long enough. The tips that I’ve just passed on to you aren’t products of my ingenuity or fruits of my knowledge. They’re some of the things I’ve been learning from my cherished friend of many years, the great pioneer American ballistician Homer S Powley. I’ve simply rewritten some material that Homer presented much more succinctly, years and years ago, in a couple of his short ballistics memos and reports. Another time, I’ll give you a run-down on an old article that he wrote ?which doesn’t exist any more, as Homer wrote it ?that’s now reflected only in other writers’ total rewriting of the information in Homer’s original manuscript ?telling how to use chronograph data to estimate chamber pressures. Stay tuned for that one.

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Copyright © 2006 Dr Kenneth E Howell. All rights reserved.

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