Accuracy is the degree of precision with which the gun can hit the target. It seems to me that there are two general sets of conditions under which shooters operate, and that the measurement of accuracy must be different under each.

“First Shot Hit” accuracy is about hitting the target with the first shot. This is the accuracy that the hunters and egg shooters are concerned with.

“Bench Rest Group Accuracy” is about shooting small groups. Small groups are the goal when shooting bench rest, or position shooting.

Jeff Brown: “Accuracy means different things to different shooters. What a deer hunter considers ‘accurate’ a bench rest shooter wouldn’t get out of bed for. Certain types of shooting require GREATER levels of ‘accuracy’ and vice versa. Consistency, the ability of the gun to perform at the same level of precision over time, is another element of accuracy.”

How to Determine “first shot hit” Accuracy

   Most rifles will shoot the first shot from a cold clean barrel into a different place than where it will center subsequent shots from a warm fouled (even slightly) barrel. If the game is to shoot one shot at a woodchuck at 150 yards, or an egg at 250 yards, or the center of the German ring target at 200 yards; then the shooter needs to know the bore condition maximizing accuracy and needs to adjust the sights for that bore condition.

   There are three possible starting bore conditions: clean and dry, clean and oiled and un-cleaned. In all cases the barrel will be cold for the first shot. (This is the least interesting testing I have ever done, I need to have another gun to shoot or I become impatient and shoot too fast-this ruins the test.)

   Use a proven accurate load for your rifle.

   Shooting is done from the bench, allowing at least five minutes between shots for the barrel to cool. This testing will take several hours. Start with a clean and dry barrel. Fire the shot. Clean and dry the barrel. Wait for the barrel to cool. Fire another shot, clean and dry the barrel. Continue. Clean the barrel with the same method each time, same number of patches, same brush if used, same cleaning solution, same everything.

   At the end you will have a set of groups, the number depending on your patience; two or three groups will tell part of the story but five-five shot groups is a good representative number. Measure the groups and calculate the average. Then do the same with a clean and lightly oiled barrel.

Then do the same with an un-cleaned barrel.

One of the three bore conditions will give you the smallest group size-for me it has most often been the un-cleaned barrel that wins. Measure the group sizes; adjust the sights, test a few more shots and you’re done. With the stated bore condition you now have the all the possible first shot error out of the gun and will hit close to center on that first shot.

Rifle bench rest group accuracy

   I have never read a discussion of rifle accuracy that defined a shooting procedure and the resulting levels of accuracy.

   In reasonable conditions, shooting cast bullets, from a good bench rest position, shooting two five shot groups and sighter’s / fouler’s at 100 yards in fifteen minutes or less, using a telescopic sight:

• -An average for five groups of five shots of between 1.5 and 2 inches is good accuracy-suitable for most offhand shooting.

• -An average between 1 and 1.5 inches is excellent accuracy- suitable for the best offhand shooting and good bench rest shooting.

• -An average of one inch or less is superior accuracy- rifle/load combinations achieving this will frequently place “in the money” at bench rest matches.

   There has been some question about these (arbitrary) definitions, to which I offer the following:

   At the 2005 CBA Nationals in Kansas City, the average 100 yard five-shot group was, for all shooters in all classes, 1.003 inches. Discounting the Long Range Handgun shooters, the rifle shooters averaged .988 inches.

        At the 2006 CBA Nationals in Oregon, for the 100 yard five shot group competitors who finished all the five shot matches and excluding the Long Range Hand gunners-all other classes-, the average five shot 100 yard group was .990".

   Lots of shooters operate in a vacuum, not knowing if their groups are good or great or what. To compound the confusion, we are often shown those “wallet groups” that are always un-representative of the rifle’s accuracy. (Why else are they carried in the wallet?)

All accuracy testing should be done from the bench. Many offhand shooters test rifles and loads shooting offhand. I believe that they would improve their loads and offhand scores by testing from the bench.

   Some shooters have written that ten shot groups are more informative than five shot groups. I disagree, and shoot five shot groups at 100 yards for most accuracy testing. The amount of information contained in one ten shot group is the same as in two five shot groups; but the opportunity for the shooter or the conditions to honk a shot out of the group doubles from five to ten shots.

   There is argument that group size measured from center to center of the widest two shots in the group is not the best measure of accuracy, and that “mean radius” or some other measure is a better measure. While I agree with this argument technically, center distance between the two widest shots is easily measured and group match targets are measured using this method, so my measure of accuracy is group size.

   I shoot smaller groups the more I shoot in a given session. Something happens as I shoot: I “settle in”, I make a series of adjustments and learn about the day. Whatever combination of things happens, they have happened to me enough to know that more shooting yields smaller groups.

        Shooting a lot of groups is necessary to tell what is going on with the rifle and load. We often see shooters come to the range and shoot one or two groups of five shots and then leave thinking that they have learned something. I wonder if they have.

How to test for accuracy

   Sometimes we want to know if a certain gun and equipment combination is accurate.

Other times we are interested in finding out if variations in powder charge or bullet hardness or primer type or bedding or scope sight or bench rest equipment or any of the other constituents of the equipment/ammunition system change accuracy. Will a change in equipment result in a change in accuracy?

To find out we need a consistent method of testing , and measuring and evaluating the results.

     I propose the following procedure for rifles and single shot rifle-like pistols:

• a. Shoot five, five shot groups at 100 yards with each set of equipment under the same conditions.

• b. See if the groups are in control. If not, back to the drawing board.

• c. Measure the group sizes to the nearest twenty-five thousandth of an inch interval, using a ruler.

• d. Average the group sizes for each set of equipment and compare them. If the larger average group size is 125% of the smaller or more, there probably is a difference. If not, there probably isn’t a difference.

     What about sights? If the same sights are used throughout the test, there is no problem.

Why five groups? Twenty-five shots with each of two lots of ammunition, plus sighters, giving the barrel time to cool between shots, seems a reasonable compromise between two little and too much, and allows testing in a period of time when weather conditions won’t change substantially-most of the time.

Why 100 yards? Shorter ranges with accurate guns yield smaller groups that may be hard to measure. Longer ranges make weather and wind conditions cause more of the variation in accuracy. 100-yard ranges are more common than longer ranges. Spotting scopes allow us to see shots at 100 yards; some shots can’t be seen at 200 yards.

To find out we need a consistent method of testing, and measuring and evaluating the results.

I propose the following procedure for rifles and single shot rifle-like pistols:

• a. Shoot five, five shot groups at 100 yards with each set of equipment under the same conditions.

• b. See if the groups are in control. If not, back to the drawing board.

• c. Measure the group sizes to the nearest twenty-five thousandth of an inch interval, using a ruler.

• d. Average the group sizes for each set of equipment and compare them. If the larger average group size is 125% of the smaller or more, there probably is a difference. If not, there probably isn’t a difference.

     What about sights? If the same sights are used throughout the test, there is no problem.

What does “In control” mean,? If the groups are not fairly round, there’s something happening that isn’t due to variations in the equipment. If there’s vertical stringing, it may be ignition problems. If there’s horizontal stringing, it may be the wind. But if the groups aren’t fairly round, they’re not in control; and until they get round, no reliable inferences can be made.

What about called flyers? My rule is that if I call a flyer-my fault-before I look through the scope, I mark and don’t count the shot.

Measuring group size

I cannot measure group sizes repeatedly to .001”. John Alexander recommends, and sent me, a plastic ruler graduated in tenths of an inch. I'm sold on this for measuring groups over 1”. This ruler can be used to measure group sizes in increments of .025", by interpolating. For example, a group size between 1.600" and 1.700" can be measured as 1.600", 1.625", 1.650", 1.675" or 1.700" by carefully looking at the ruler and the target, and mentally dividing the ruler interval into four parts. I must have good bright light and occasional use of a magnifier to use this ruler graduated in tenths of an inch.

I measure small groups with a dial caliper under magnification and I keep in mind that the third digit to the right of the decimal is fuzzy.

Ken Mollohan: It’s true enough that the average person will have trouble measuring hole spacing in paper to 0.001” reproducibly with calipers, but measuring to 0.01” repeatedly isn’t at all difficult. Dial calipers are readily available for about $20 or $25. Simply lay the target on a flat, hard surface, and set one contact surface at the edge of one hole, and move the other surface until it is on the same edge of the second hole. Don’t even try to measure centers: The distance from the eastern edges of two holes will be the same as the distance between the centers of the holes, and much easier to locate visually. Also, most scientific supply stores sell inexpensive optical magnifiers that are simply set on the surface of a target. A set of lines are etched on the lens that looks a lot like a ruler when viewed through the lens. Simply set the zero point at the edge of one hole, and read the distance directly off of the scale. Almost any desired degree of accuracy is obtainable, depending on the magnification and the etched scale you select. Many of these magnifiers are only about an inch or less in diameter, but I believe they are made up to about three inches in diameter.)

Group Sizes And Statistics

   (Thanks to Brent Danielson, Pete Mink and John Alexander for their editorial assistance. Thanks to Jeroen Hogema of The Netherlands for his statistical contributions. I claim exclusive credit for all errors.)

   In my mind the primary function of Statistics is to translate sets of numbers into words-into sentences or statements that we all can understand.

   When we shoot groups of shots and measure group sizes, these group sizes are the sets of numbers; now we’re going to find out what statements can be made.

Definition: We shoot with a certain gun and scope and bench rest and bullet and powder and primer and the whole megillah. I’ll call his whole set of stuff a “load”, and we’ll talk about how a “load” shoots.

Estimating Group Size (This is about 5 shot groups)

   There is some “inherent” or “long-run” or “true” average group size for each load.

   Each group we shoot is an estimate of that long-run average. When we shoot sets of, for example, five five-shot groups; the average of the five group sizes is an estimate of that long-run average.

   We can never know exactly what the long-run average group size for a given load is.

We can calculate, for various levels of certainty, the bounds within which that long-run average exists.

We can be 90% sure, or 99% sure, or any percent sure of our bounds. We select how sure we want to be.

The surer we are, the broader the bounds of that long-run average.

Here’s a table of bounds, at 95% sure and 99% sure for group averages where the number of groups varies from 2 to 30. See the bold entries at Number of Groups, 5. The 95% sure bounds are at 133% and 67%. We can make a statement, thus: “We are 95% sure that the long-run average group size is between 133% and 67% of the average of five group sizes.”

See also that the 99% sure bounds are at 155% and 45%. We can make another statement thus: “We are 99% sure that the long-run average group size is between 155% and 45% of the average of five group sizes.

     So we see the bounds broaden as we get more sure. We’ve got bounds, we’ve got sure. The more precise one is, the more vague the other.

And we see that the bounds narrow as the number of groups averaged increases, although the narrowing slows pretty much by the bottom line, 30 groups.

This table and the accompanying graph give me a notion of what we know when we shoot a set of groups and measure and average the group sizes. Precise calculations using this table information is both a little fuzzy for some boring reasons, and is not of much or any value that I can find.

That said, here’s how to calculate and construct statements:

   On 26 January, 2005, with a M54 Winchester in 30 WCF, five 5-shot groups averaged 1.265 inches.

          Back to the statements:

“We are 95% sure that the long-run average group size is between 133% and 67% of the average of five group sizes.”

133% of 1.265 inches = 1.682” / 67% of 1.265 inches = .848”.

   The statement becomes: “We are 95% sure that the long-run average group size is between 1.682” and .848” when the average of five group sizes is 1.265”.

And, “We are 99% sure that the long-run average group size is between 155% and 45% of the average of five group sizes.

155% of 1.265 inches = 1.961”

45% of 1.265 inches = .569”

The statement becomes: “We are 99% sure that the long-run average group size is between 1.961” and .569” when the average of five group sizes is 1.265”.

For the source of the table and graph see “small sample estimator of confidence interval for mu.xls” , an EXCEL workbook, in the Appendix.

Group Size Variation

John Alexander triggered this analysis with this observation, from an e-mail:

"... Several years ago somebody pointed out that more variation than you might think was natural.  By way of illustration they pointed out that there is huge amount of data for five five shot groups in all the "Dope Bag" tests and that the largest group of the five averages very close to twice the smallest group.  I went right to work with a couple of years worth of Am Rifleman (maybe 30 guns tested and perhaps three types of ammo for each) and was astonished at how consistently that held true. Not only the overall average but the biggest and littlest groups in individual five five shot groups were often very close to a ratio of 1:2. It was spooky how often..."

 For 5-shot and 10-shot groups, the table shows the relationship between the number of groups fired and the ratio between the largest and smallest.

   Note that for five groups the table tells us to expect that the ratio of the largest group to the smallest group will be 1.91:1. This corresponds to John Alexander's observation of a ratio of 2:1.

   So, for example, if you go to the range and shoot five 5-shot groups, then the largest should be about twice (1.91 times) the size of the smallest-on the average. If you shoot seven 5-shot groups, expect the largest to be 2.14 times the smallest. Not every time, this will vary from day to day, but on average expect these amounts of variation.

    What does it mean?

   If your group sizes vary about as the table predicts, your shooting is as would be expected.

   If your group sizes vary repeatedly and substantially more than the table predicts, look for something unusual in the load. I would suspect that in this case that the average group size would be large. Something is out of control, and affecting accuracy in an unusual way.

   If your group sizes vary repeatedly and substantially less than the table predicts, I'd like to hear from you. You're doing something good that we need to learn about.

     Here's the next paragraph from John Alexander's e-mail:"...  today I went down in the basement and did the same thing with a dozen copies of recent Am Rifleman Dope Bags. I didn't take the trouble to average the ratios but at least two thirds and maybe three fourths of the ratios were less than 1:2, most were much less with many of the ratios closer to 1:1.2.  I wonder if the new folks running the tests are throwing out a few "fliers"..."

The 2002 and 2004 CBA National Match records were reviewed and analyzed. These matches include four 5-shot groups at 100, then 200, yards. The average for the Largest/Smallest group sizes at 100 yards was 1.83, at 200 yards it was 1.87; these against the table estimate of 1.77.

For 10-shot groups, two of which were shot at 100 and 200 yards, the average for the Largest/Smallest group sizes at 100 yards was 1.346, at 200 yards it was 1.31; these against the table estimate of 1.25.

Close enough for government work.

For an explanation of the table, see “estimating group size variation .xls”, an EXCEL workbook in the Appendix.

Detecting Accuracy Differences

   We sometimes wish to know if one load produces smaller groups than another. If Remington 2 1/2 primers shoot into 1.2" for five 5-shot groups, and if Winchester WLP primers shoot into 1.1" for five 5-shot groups-all other things being equal-do I know that the WLR primers will produce smaller groups? Sometimes the same test on a different day yields opposite results.

   If one load shoots five shots into half an inch at 100 yards, and another load shoots into eight inches; there's no problem figuring out if there is a difference. We might want to shoot another pair of groups to check, but if the difference between group sizes is very large, the decision is easy. Difficulties arise when two loads shoot into groups that aren't very different. Then deciding if one load shoots smaller groups than another is tougher.

The Wilcoxon Rank Sum Test

The easiest way I know of to apply statistics to two sets of group measurements, (or any two sets of numbers), is the Wilcoxon Rank Sum Test. This test doesn't require a computer or spreadsheet or any fancy footwork, and doesn’t require the (problematical?) assumptions that t or Z tests include.

This test answers the question: “Given two sets of group sizes, are the two sets from the same long-run average group size? (Or, are the two sets of group sizes from different long-run average group sizes?)

Here's an example of how to do the test:

We have two different loads and want to know if there is a difference in their accuracy. We shoot groups with each load, load "A" and load "B", measure the groups, and write the group sizes down in columns headed "A" and "B".

Then we "Rank" the group sizes. See that 1.62 is the smallest group in either column and is ranked "1". And 1.71 is the next smallest, ranked "2", and so on. 2.71 is the 11th ranked group in the two columns. (In the case of a tie, use the average rank. Ex: The 4^th and 5^th ranked number are the same. Use a rank of 4.5 for each.)

When they're all ranked and checked for correctness, we add the ranks in each column. The total of ranks for "A" is 25, and for "B" is 66.  Call these the "rank sums".

(If there is a "tie", if , for example there are two identical values that would be in third and fourth place, then assign each a rank of 3.5.)

There are 6 “A” samples and 7 “B” samples, we say that for “A”, n = 6 and for “B”, n = 7.

     We'll use both rank sums, 25 and 66. Now go to the table.

Look at the intersection of n = 6 and n = 7, see the numbers 28 and 56. Since 25 is not equal to or between 28 and 56, and since 66 is not equal to or between 28 and 56, we reject the notion that the two sets of group measurements are from the same long-run group average. And we conclude that there is a difference between the accuracy of loads A and B.

To get into the weeds a bit:

     The statement might be: “Based on the rank sums shown above, we are not 95% sure that the groups came from the same long-run group average.” If the rank sum of one of the set of groups had been between 28 and 56, the statement might be: “Based on the rank sums shown above, we are 95% sure that the groups came from the same long-run group average.”

   See the Appendix for the “Wilcoxon Rank Sum Worksheet” that may be copied and used for your analysis.

Is load “A” different from load “B”?  The “t” test

On the CD there is an EXCEL workbook called “t testing.xls”. This workbook shows another method of testing for a difference between loads.