Non-Velocity Use Of Standard Deviation


Junior Doughty

While researching my FRUGAL OUTDOORSMAN article about homemade round ball patches, I observed that the Lee .311 cast round ball shot well with my homemade patches. In fact, with the homemade patches, it shot as well as the Hornady .310 swaged round ball. That got my mind to wondering, to paraphrase Lightnin' Hopkins.

In the first place, I had sorted those cast balls via a hasty visual inspection as I cast them, not by weight or by careful visual inspection after they cooled. Some of them surely had external or internal defects. They were also cast from an unknown alloy, not pure lead. In the second place, the factory-made balls were swaged from pure lead. They should be perfect, right? They should shoot much better than the Junior-made balls, right? Wrong. They didn't. So why?

I decided to take a random sample of 10 balls of both types and weigh them, checking for weight variations. Here's the results:

Factory-Made .310 ball

Junior-Made .311 ball


45.1 45.2
45.0 45.0
45.1 44.8
45.3 45.0
45.4 45.2
45.3 45.0
45.2 45.1
45.3 45.2
45.0 44.9

Total = 451.7

Total = 450.4

Average = 45.17

Average = 45.04

Extreme Spread =       .4

Extreme Spread =       .4

The first thing I noticed was that the .311 ball was lighter than the smaller .310 ball. That verified that it was not pure lead.

The second thing I noticed was that, as I had expected, the Junior-made ball seemed to have greater weight variations than the factory-made ball in spite of their identical Extreme Spread, ES, of .4 grain. Or, I wondered, did it seem that way because I expected that result and I see some 9s and 8s in the data?

Well, I knew how to answer that question—compute the Standard Deviation of both groups.

I opened the Spread Sheet section of MicroSoft Works. After a couple of hours of figuring and entering and head-scratching, I discovered that one of the spread sheet's included mathematical functions computed the Standard Deviation of a range of cells. In just a few minutes, I had my answer.

Factory-made .310 ball Junior-Made .311 ball
Standard Deviation =   .142 Standard Deviation =   .128

At the end of this article, you will find instructions on how to download—for free—a copy of the MS Works spread sheet used in this article and containing a Standard Deviation section, a Taylor's Knockout Value section, and a Bullet Energy section.
Now, that was interesting. In spite of the fact that the .311 data appeared to show that its weight varied more than the .310 data, it actually varied less.

Hummmm, I told myself, this Standard Deviation stuff is useful for something other than velocity, statistics, classroom grade curves, and sociological/psychological surveys.

Standard Deviation is just a way to determine a variable's deviation from the mean, the average. Let's say, for example, you think you're too kinky so you go to a psychologist and take a test to determine your level of kinkiness. You answer all the weird questions; the psychologist grades the test; then you get a score, a kinkiness score. (Your deviation level?) He/she (pardon the pun) then compares your score with the test's average and its Standard Deviation. Then he/she says, "Mr. Jones, you are three Standard Deviations below the mean."

You say, "Oh, my God! You're not going to have me arrested, are you?"

He/she says, "Of course not. Jail is no place for a prude like you."

In the case of my .311 ball's score:
    One Standard Deviation = .128 grs.
    Two Standard Deviations = .128 x 2 = .256 grs.
    Three Standard Deviations = .128 x 3 = .384 grs.

So if one of the .311 balls weighs three Standard Deviations below the mean of 45.04 grs, it then weighs/scores: 45.04 - .384 = 44.656 grs.

Got that?

The Standard Deviation (SD) bell curve tells me that:
    68% of my .311 balls will weigh within ± .128 grs of 45.04 grs.
    95% of my .311 balls will weigh within ± .256 grs of 45.04 grs.
    99.7% of my .311 balls will weigh within ± .384 grs of 45.04 grs.

To explain Standard Deviation in terms of velocity, my Lyman Great Plains Hunter shoots the Lee 320 gr R.E.A.L. bullet at an average velocity of 1262 fps and with a Standard Deviation of 13.

    One Standard Deviation = 13 fps.
    Two Standard Deviations = 13 x 2 = 26 fps.
    Three Standard Deviations = 13 x 3 = 39 fps.

    The velocity of 68% of my shots will be from 1249 — 1275 fps.
    The velocity of 95% of my shots will be from 1236 — 1288 fps.
    The velocity of 98.7% of my shots will be from 1223 — 1301 fps.

When looking at the above, remember, in the 68% line for example, that the fps predictions are for one Standard Deviation below the average of 1262 fps and one Standard Deviation above it, etc., etc. 1249 +13 + 13 = 1275. Got it?

If you got it, you'll surely get the following. It's a great non-velocity use for Standard Deviation.

Many cast bullet shooters, including this one, weigh every bullet, especially for competitive shooting. The problem arises when you start trying to sort a batch of weighed bullets. You average their weight, of course, but how do you determine if a bullet is too heavy or too light? In other words, heavier than what and lighter than what?

The answer is simple. You begin by finding the Standard Deviation of the batch of bullets. You then let the Standard Deviation tell you if a bullet is too heavy or too light.

The only thing you'll have to decide is whether to sort them by one, two, or three Standard Deviations, i.e., low, medium, or high variation. Pay attention to the following, and you'll also learn how to sort a batch of bullets and actually weigh only 10 of them.

320 gr Lee .50 caliber
R.E.A.L. bullet
Average = 327.33
ES = 2.8
SD = .849
From a batch of 70, 320 gr Lee .50 caliber R.E.A.L. bullets cast of pure lead, I selected 10 bullets at random and weighed them, then entered the weight data in my spread sheet. You see the results on the right.

From the results we see that:
    One Standard Deviation = .849 grs.
    Two SDs = .849 x 2 = 1.698 grs.
    Three SDs = .849 x 3 = 2.547 grs.

    Average weight = 327.33 grs.

Mainly because I intended to shoot some 150 yard groups with those bullets, I decided to be picky and sort them by one Standard Deviation: 327.33 grs ± .849 grs.

    High weight = 327.33 + .849 = 328.179 = rounded = 328.2 grs.
    Low weight   = 327.33 - .849 = 326.481 = rounded = 326.5 grs.

I then set my RCBS powder scale to exactly 326.5 grs, the low weight value. One by one I started gently placing bullets into the pan and observing the pointer and the beam. I was not interested at all in weighing the bullets. All I wanted to know was did each bullet weigh at least 326.5 grs. Some did not raise the pointer at all. They went into the Too Light Pile. Some raised the pointer at or near the zero mark. If at or above the zero mark, they weighed 326.5 grs or more so they went into the Maybe Pile. Below the zero mark, even as little as .1 gr below, they weighed less than 326.5 grs so they went into the Too Light Pile. Most of them immediately raised the pointer to its upper limit. They went into the Maybe Pile.

All of the bullets having then been checked for maximum low weight, the score stood:

    Too Light Pile = 8
    Maybe Pile = 62

I then set my RCBS powder scale to exactly 328.2 grs, the high weight value. Again, one by one I started gently placing bullets into the pan and observing the pointer and the beam. If they did not raise the pointer or raised it to or below the zero mark, they weighed 328.2 grs or less so they went into the Good Pile. If they raised the pointer above the zero mark, they weighed more than 328.2 grs so they went into the Too Heavy Pile.

In a couple of minutes, the score stood:

    Too Light Pile = 8
    Too Heavy Pile = 10
    Good Pile = 52

And my batch of bullets were sorted by one Standard Deviation: 327.33 grs ± .849 grs.

Sorting bullets by weight
  Too Heavy Pile on the left. Good Pile in the center. Too Light Pile on the right. That's my college Psych 1010 textbook beneath the scale. Where do you think I learned about kinkiness tests?
Looking closely at the results, we see that 52 of my 70 bullets, 74% of them, fell within one Standard Deviation of the average. That is very close to the bell curve's prediction of 68% or 48. That proves to me that computing the Standard Deviation of a batch of cast bullets is the way to sort them accurately.

My weigh by Too Light and then by Too Heavy method is very fast. It took me less than 20 minutes from the time I started weighing the 10 random data bullets until the last bullet in the Maybe Pile was checked for Too Heavy.

I have individually weighed my last batch of cast bullets.

You can download, for free, the exact same spread sheet I used in this article.

First, it is a MicroSoft Works spread sheet, so if your computer doesn't have MicroSoft Works or MicroSoft Excel installed, don't try to download the Standard Deviation spread sheet because you can't open it.

Got that? Don't download the file if you ain't got the program to open it. When you do download it, save it where you can find it, ok?

Second, if you download the file and open it, don't enter data in any cells except the cells within the red-bordered boxes. You might accidently erase a formula cell. If you can't follow directions and screw up, delete the file from your computer and download it again.

Note: If you own a chronograph, the spread sheet will give you a slightly different Standard Deviation than the chronograph due to a slight difference in the formulas used.
The spread sheet contains four sections. The first section, the top one, computes Standard Deviation for up to 15 variables, i.e., data entries. If I was trying to get Standard Deviation for a batch of more than 150 bullets, say 200 to 500, I'd use 15 entries. For 150 or less, I'd use 10 entries.

The second section computes Taylor's Knockout Value for a bullet. Bullet #1 in the spread sheet is my .32 caliber muzzleloading squirrel rifle bullet. Bullet #2 is my .22 caliber rimfire squirrel rifle bullet. The table explains why my .32 kills squirrels as or more effectively than my .22 in spite of the fact that the velocity of the .32 is much lower than the .22.

The third section computes a bullet's energy.

The fourth section computes a gun's recoil energy. If your gun kicks too hard and you're thinking of trying a lighter bullet, use this section to compare the gun's recoil energy with both bullets.

All four sections will compute up to seven values. For example, like I did with the Knockout values for my .32 and my .22, you can compute the values side-by-side and easily compare them.

Now, here comes the download instructions. If you can't do it, ask your kids to help. Don't ask me.

The directory on contains a file named formula.xlr and a file named

formula.xlr is the raw Works spread sheet. is the raw Works spread sheet compressed into a zip file.

If you are using Internet Explorer, not Netscape, as a browser, right click on formula.xlr Then left click on "Save Target As." When the little "Save As" window opens on your computer, enter the extension ".xlr" after the filename "formula" in the little box. If you don't do that, Windows will save the file as formula.html, and you probably can't open it. If you do that, Windows will save the file as formula.xlr and you're in business.

If you can't simply right click and download the file, then left click on and download the zip file and unzip it with whatever and however your computer unzips files.

Remember: don't ask me. Ask your kids.

Remember: don't try to download either file unless your computer has Works or Excel installed.

Ready? Click here:

Copyright 2001 by Junior Doughty

HOME   Back To The Shooting Section