Statistics

# Probability And Shot Groupings

That’s some mighty fine shootin’, son.

I was reading the WikiPedia page on “accurizing” firearms, and came across a claim I find suspicious:

Statistical likelihood says the fewer shots that are fired, the smaller the dispersion will be. 3 or 5-shot groups are acceptable for zeroing the sights and rough accuracy estimates, but most shooters consider 10-shot groups to be the minimum for accuracy comparisons.

For standardization, it is best to fire five-shot groups with the same aiming point. It is a statistical fact that group size will increase with the number of shots fired.

If I’ve learned anything from reading wmbriggs.com (and I do try), it is that probability distributions quantify our uncertainty about a future event, rather than describing how things would look if we could observe an infinite number of those events. In other words, we expect there’s a better chance that the next shot will be close to the last, rather than far (relatively speaking). But there’s still a chance that the second shot fired could be an outlier, while the 48 shots after it could all land on the same hole. Is there any reason to believe that the more shots we fire, the better idea we’ll have of the “true” group size?

First, I think we can all agree that accurizing is one of the worst gerunds since concretizing: it should be shot and put out of its misery.

Second, group size. As defined by Sniper Country: “It is the maximum distance between the centres of the two farthest shots in a group. The easiest way to do this is to measure from the outside edge of one bullet hole to the inside edge of the farthest one away.”

The claim “It is a statistical fact that group size will increase with the number of shots fired” is true. Fire one shot. What’s the group size? Zero inches. Fire a second shot. Unless you Robin Hood it (and there is a chance this happens) the group size for both shots will be larger than the group size for just one shot. The more shots you fire the greater the chance the rounds wander from their appointed path, and thus the larger the group size.

Probability does, as you say, quantify our uncertainty about future (or really any) events, even in cases where we have less than infinite information—which we never do. And experience does show that successive shots will be closer to one another than shots taken at widely different times.

It’s your last question that is of most interest: “Is there any reason to believe that the more shots we fire, the better idea we’ll have of the ‘true’ group size?” There is no such thing as true group size, but the more shots fired under a set of conditions the better we can characterize our uncertainty in what the group size will be.

There are sets of conditions; premises, if you like (to fit in with our usual language) which describe the “state of firing”. These includes physical conditions like (of course) weapon type, ammunition, position, distance to target, even the weather. But it also includes many intangibles such as your mood, blood alcohol content, health, and your interest in hitting the target, which might include noting whether the target is firing back at you, and so on.

Take a set of conditions and a number of rounds fired and marry that to some historical performance measurements and we could in theory calculate the probability (maybe only crudely) that the group size will equal some specific number, or that it will be between two given numbers. The number of rounds fired is crucial, because as we’ve already seen, the more shots the larger the group size usually.

This is different than actually measuring group size under a set of pre-specified physical conditions, as in competitions. There, the average of some number groups is taken, say 10 rounds of 5 shots, maybe varying the physical conditions slightly between rounds. The smallest average wins. Whatever your average is can be used as information to calculate the chance of future group sizes measurements.

Here’s some fun homework. The media, God bless them, often make much of the number of shots fired by policemen when they’re trying to take down a bad guy. Suppose the physical conditions include a bad guy firing back at the policeman. Further suppose that we know for a fact that each round the policeman fires has a 0.3 chance of hitting its target and incapacitating the bad guy. How many shots does the policeman have to fire so that there is at least a 99% chance of bringing down his target? The answer will surprise you and (probably not) delight your media friends.

Hint: if he fires only once, he has a 30% chance of bringing down his target. He does not have a 60% chance if he fires twice.

Categories: Statistics

### 42 replies »

1. Will says:

I think maybe the bore-sighting folk mean that after 10 shots the firing conditions have changed enough so as to make further testing less effective. The barrel heats up, for example, and the gun may shift in its stand (if one were used).

Regarding the police shooting; there is no way to ensure 99% success rate give the conditions described.

2. MattS says:

Briggs,

“The media, God bless them, often make much of the number of shots fired by policemen when theyâ€™re trying to take down a bad guy. Suppose the physical conditions include a bad guy firing back at the policeman.”

How many cases have been in the news over the last couple of decades from New York and LA where multiple cops have emptied their weapons on “suspects” that turned out to be unarmed? A dozen, more?

“Further suppose that we know for a fact that each round the policeman fires has a 0.3 chance of hitting its target and incapacitating the bad guy.”

Bad assumption, of the cases I mentioned above there are several with totals of 40 or more rounds fired and the “suspect” was hit fewer than 10 times.

If a cop has to fire 10 – 50 rounds to hit a suspect once or twice, how many cops will it take to shoot their way out of a paper bag?

3. MattS says:

That should be 10 – 15 in the final question above.

4. Ken says:

Nice summary…but…incomplete on engineering factors (or, too stats-heavy):

– A few shots vs. many shots to zero a gun sight (achieve a smaller grouping):

Fewer is generally better — after a few (around five-ish) fired from a gun at ambient temperature the barrel heats up enough to alter the apparent center of a shot group enough to be a factor for anyone involved in competitive shooting. Here’s one ‘analysis’ of that (see if it matters to your situation): http://www.stocks-rifle.com/barmovpg2.htm . For a sniper or competitive shooter [where the entire rifle stock is made of materials designed to hold shape regardless of temperature variations] a few millimeters makes a difference; for most of us, not worth the bother to fuss over. In either case, this is really not an issue warranting statistical analysis.

Type of round used matters a lot more–some brands have more manufacturing variability than others (bullet weight & contour, and, powder energy being the most noteworthy–causing greater/lesser group sizes due to more varied bullet speed & trajectories.

– Policeman example — 0.3 chance of bringing down the bad guy:

It might be a nice statistical exercise giving some real insight into what really happens…but….Any practical probability analysis of shooting needs to account for the type of weapon used & other factors. A number of myths need to be understood, including the curious fact that a truly objective analysis of real-world shootings does not exist — they’re that variable & pertinent details are not always known.

FBI has addressed these & other factors in a short report: http://www.firearmstactical.com/pdf/fbi-hwfe.pdf

For example: In the time it would take the average person to draw, aim and fire a handgun (approximately 1.5 seconds), an average assailant could cover about 20 feet — that’s an unexpected blitz attack from the end of a typical driveway where the shooter was alert to expect the attack but did not respond until the assailant makes the first move! After a fatal shot (e.g. thru the heart) fired on someone who unexpectedly attacked in such manner (such a shot is highly unlikely in real-world circumstances by anyone), one should expect that before a second shot could be fired the attacker would be upon you (if you stood still). If the attacker had a knife they would very likely succeed in stabbing you, quite possibly fatally and/or more than once (stabbed more than once and/or stabbed with multiple would two or more of which would be fatal to you — by a person bleeding to death with no hope of surviving that bullet wound).

This is a drill often done, short of actual shooting/stabbing, in firearms safety classes to illustrate how far away a blitz attack can commence vs. how long it takes to draw a weapon, much less fire it accurately — in many such examples the student is unable to react [even knowing what the game is about] in time to raise the gun barrel to horizontal before the instructor/assailant is upon the student & in a position to hug & shove the student).

Take this into consideration the next time you hear the accusation that “the police continued to fire.”

Also, be aware that most police (excepting SWAT & other elite units) get to the range only annually to qualify — a lot of hobbyists shoot much more and are even better than law enforcement. Which is to say the vast majority of gun-toting individuals (law enforcement & conceal carry permitted) aren’t all that great shots, many are quite lousy….and they’re all much worse under stress of real combat.

To understand the psycho-behavioral effects under stress see: http://www.killology.com, or if at all interested, read: On Killing: The Psychological Cost of Learning to Kill in War and Society, by D. Grossman.

5. MattS says:

“Take this into consideration the next time you hear the accusation that â€œthe police continued to fire.â€

Also, be aware that most police (excepting SWAT & other elite units) get to the range only annually to qualify â€” a lot of hobbyists shoot much more and are even better than law enforcement. Which is to say the vast majority of gun-toting individuals (law enforcement & conceal carry permitted) arenâ€™t all that great shots, many are quite lousyâ€¦.and theyâ€™re all much worse under stress of real combat.”

Sorry, I think it’s perfectly reasonable to expect the police to be better shots than most currently are and that cops that aren’t aught to be taken off the streets and forced to re-qualify.

6. Sheri says:

MattS–Most cities can’t get by with just a couple of cops for extended periods of time. Hope you’re going to stagger the re-qualifying.

7. Gary says:

“… even in cases where we have less than infinite information â€” which we never do.”

I don’t know about you, but I ALWAYS have less than infinite information about everything. Makes decisions so much harder. Can I borrow some of that infinite information next time I’m wrestling with a decision? 😉

8. DAV says:

MattS,

There’s a lot behind that “stress of combat”. One needs a lot of nerve to take steady aim when the other guy is shooting back. The tendency for wild, quick shots is enormous. Some time ago there was a dash cam video of a shootout between a state trooper and (I guess) a bad guy across the hood of a car. Neither one of them got hit. The trend for using higher powered ammunition (magnums, +P, etc.) does little to keep one’s arm focused on the target.

9. Bob Koss says:

13

10. Ray says:

That’s the same problem I had in an operations research course, only it was about anti aircraft missiles. The missile has a probability of 75% of obtaining a kill. How many missiles do you shoot at the target to obtain a 95% kill probability? You have to work the problem in reverse and calculate the probability of not obtaining a kill.

BTW,in the Vietnam war I saw estimates of 3000 bullets fired to kill one person.

11. DAV says:

Ray,

Not the same thing. Sounds like a bulletsUsed / enemyKilled ratio, In combat, most of the fire is to provide an incentive to not move or make it harder to do so.

12. MattS says:

Dav,

“Thereâ€™s a lot behind that â€œstress of combatâ€.” I am not claiming that there isn’t. However, those who can’t keep a cool head and shoot straight even under that stress shouldn’t be cops.

Sheri,

“Most cities canâ€™t get by with just a couple of cops for extended periods of time. Hope youâ€™re going to stagger the re-qualifying.”

They should have done a better job qualifying them in the first place.

The city with the largest known issue of cops who can’t shoot straight is New York, LA would be second.

New York has a ratio of LEOs to total population almost an order of magnitude higher than the national average.

Sorry, I have no sympathy for them.

13. Mike Ozanne says:

Surely, with the Cop example, assuming return fire and tactical move etc, each shot is independent and the answer is, until success or you run out of clips, whichever comes first…

14. HowardW says:

MattS

Few cops fire weapon in anger during career, so there is no way to know if they will be a cool headed shooter. Also, the cool shooter is not necessarily related to how well they can hit paper targets in a relaxed, safe setting. The movie Unforgiven had a scene that sums it up well.

Little Bill Daggett: Look son, being a good shot, being quick with a pistol, that don’t do no harm, but it don’t mean much next to being cool-headed. A man who will keep his head and not get rattled under fire, like as not, he’ll kill ya. It ain’t so easy to shoot a man anyhow, especially if the son-of-a-bitch is shootin’ back at you.

If Police are poorly screened and trained and don’t maintain skills, then one would expect inaccurate and irresponsible shooting. They don’t make the program, they just have to perform based on the consequences. It’s too easy to love and hate cops.

You bring up some good points, but could use some empathy. It’s a thankless job with an early retirement and a young death from stress and alcohol induced coronary artery disease.

Three are good enough. After 5 shots, the shooter begins to tire. Besides, you need to make windage and elevation adjustments then shoot more, so it could take a while to get sighted in. More than 5 each adjustment would be too taxing. Even shooting from a rest. The Marine recruits spend a whole week dry-aiming at fake targets painted on barrels just to get used to holding the position where you make a rest using your bones, not muscle. Snapping in they call it. It’s kinda like yoga.

I don’t know the statistical answer, but to be 99% sure with a dangerous target, all you have is not enough.

15. MattS says:

Howard.

Yes, the job of a cop is difficult, but those who can’t handle it are best advised to seek a different career. Coddling those who can’t handle the job so the can limp along in a law enforcement career is not empathy in my book, it’s insanity.

Maintaining the necessary skills should be part of the job and failing to do so should be grounds for termination.

My lack of sympathy expressed above is not for the individual officers, but for the departments that might have to find themselves re-training a large percentage of their work force.

16. DAV says:

MattS,

No one knows how they’ll react under fire until they’ve been there so expecting an organization to weed out those that would panic is unrealistic.

I think the real problem is we are getting too many cops who are trigger happy and are likely to get free passes. This, too, is something hard to test for.

17. Briggs says:

Bob Koss,

How’d you come to that number? No credit without showing your work.

All,

Of course the example is contrived, as the text indicates. There’s no practical way to know in any individual situation. But the homework is illuminating. Whoever gets the complete right answer should email Bruce Springsteen

18. 0.3 = chance of hitting target.

0.7 = chance of not hitting target.

1 – (0.7)^n = 0.99

at which point I didn’t have my calculator handy and put the left side into excel and typed numbers in….

which isn’t quite true.

I put 1 – 0.3^b1 in and got 4 for an answer…..

stuttered at my stupidity and revisited my premises and inversion analysis…

Then I read Kens answer and liked it every so much more. 13 is a nice number and all, but reality is ever so much more important.

What is the ratio of bullets fired on television TV Crime shows in one night to bullets fired by the entire US police force in non training situations for a year?

19. Sheri says:

MattS: Just stating a fact, not looking for sympathy for the city or the cops.

There are training methods that can simulate being under fire quite effectively. I’m not saying they perfectly simulate being under fire–nothing can do that. However, some of these are effective enough to have participants realize just how easily one can die if one does not pay attention and remain on focus. The FBI trains hostage rescue teams with live-fire exercises. It’s all expensive which one supposes is the reason the resources are not widely utilized by police departments. (MattS: You are correct that no one can weed out everyone who will panic under fire, but many can be identified before they are turned loose in public.)

20. Briggs says:

You have hit upon the well known fact that Excel stinks at math. But you’re almost there.

21. Neither money nor time are unlimited resources. It would be wonderful if all cops were better trained in shooting, but considering how seldom cops actually use thir guns, it isn’t surprising that training for an unlikely occurance is low on the priority list. Cops also need training in driving, in the law, in the ever-changing paperwork requirements, in all sorts of things.

22. dr.bill says:

Bob Koss is correct. Here’s the general solution:

p = probability of a hit on a single shot
(1-p) = probabliltiy of a miss on a single shot

p + p(1-p) + p(1-p)Â² + … + p(1-p)â¿ = cumulative probability after n+1 shots

P = cumulative probability after n shots = 1 – (1-p)â¿

Solve for n after choosing p and P.
Round up to the next integer.

23. Briggs says:

Dr B,

I give versions of this to my students. They are always amazed that the number is so high.

24. Scotian says:

Brad, you don’t need excel or a stand alone calculator if you have google. Just type in the word calculator and solve for -2/log(0.7). This gives Bob’s number. You’re a hard grader Briggs, no inflation in your courses.

25. MattS says:

Dav,

“No one knows how theyâ€™ll react under fire until theyâ€™ve been there so expecting an organization to weed out those that would panic is unrealistic.”

You may well be correct that weeding them out ahead of time is unrealistic, though I don’t accept that that as a given.

“I think the real problem is we are getting too many cops who are trigger happy and are likely to get free passes. This, too, is something hard to test for.”

I disagree with this on several points.

Hard to test for does not equal to impossible to test for. Even an imperfect test would result in fewer trigger happy cops. Given the consequences it’s worth the effort.

The real problem is more complicated.

First, Those mentally unfit for the job, both those who panic and those who are trigger happy are far to difficult to get rid of once something bad happens as a result. The best thing for both those officers and the public is to sit them down and tell them to find another line of work.

The size of a police force greatly effects the ability to weed out the unfit, and probably not in the way you think. The larger the size of the police force relative to overall population the harder it will be to weed out the unfit. The reason for this is they can’t meet the manpower requirements with strict fitness requirements. Does NYC really need a police force that is 10 times the size of the national average on a per capita basis?

The militarization of police generally not only directly encourages the trigger happy, it creates pressure towards ever larger police forces.

26. Bob Koss says:

I know the formula due to it being similar to calculating some of the odds when playing poker. It is (1 – .3) ^ 13 = .0097

Do I get a cookie now?

27. Bob Koss says:

That is the probability of not hitting the target.

28. Uncle Mike says:

Easy problem. 13 is right. Here’s a similar one from the real world:

The USFWS reports that since they took over “protecting” the Northern Spotted Owl in 1989, the NSO population has declined 3 percent per year. What percentage of owls are alive today as compared to the original population?

Bonus question: The failed program to “save” the NSO has cost the region \$10 billion per year in lost economic activity. What is the total cost to date?

29. Did you take into account that a cold barrel is more accurate than a hot barrel? Old Marine talking here.

30. Bob Koss says:

Uncle,
1989-2013 inclusive would be 25 years.
.97 ^ 25 = .467 remaining.
Total wasted = \$250 billion

31. Bob Koss says:

Here is a common poker problem.

You are playing 6-handed 7-card stud where everyone starts with 2 down cards and 1 exposed card. You hold 3 hearts. No other player shows a heart. At the end of the betting round 3 of the other players have dropped out. The remaining players get their 4th card. Your card is another heart, but one other remaining player has also received a heart.

At this point what is the probability of you receiving at least one more heart to make a flush by the 7th card?

32. Rich says:

n = log(1-P)/log(1-p) For P=0.99 and p=0.3, n=12.9. Round up to 13. Mathematics provides so many ways to do the same thing.

“Robin Hood it”! and you complained about accurized!?

33. Milton Hathaway says:

The “group size” of the shots seems like focusing on the tails of a normal distribution. Certainly not something I do much in my work, but I can see that when shooting real bullets, “ignore the outliers” may not be the best advice.

As far as the 99% incapacitate problem, I solved the wrong problem – I didn’t scroll back, and remembered the cop’s hit rate as 40% per shot, which gives a slightly more interesting answer: 9 shots for an engineer, 10 shots for a mathematician. (I haven’t a clue what it would be for a statistician.)

Not to brag, but Martin Gardner taught me to figure out these types of probability problems when I was just a young kid. Basically it’s a word problem. If you state the problem using the word “or”, you might fool yourself into thinking that you can add up the individual probabilities, as in “cop hits the first shot or the cop hits the second shot or . . .”. Instead, restate the problem using word “and”, then multiply the individual probabilities, as in “cops misses the first shot and cop misses then second shot and . . .”.

As far as some of you anti-cop types, if your fellow citizens in your city feel as you do, I’m sorely tempted to say you just might deserve the police force you will invariably end up with.

34. revGDright says:

I spend time over the holidays working up some new reloads and have 2 lots of ammunition, A and B, and I want to compare the accuracy of them. I get my Remington and drive the 30+ miles out to the nearest rifle range and set up on a bench rest to start shooting at 200 yards. After some clearing shots, I fire 10 rounds of ammunition A at the target and get a group size of 3 inches (A great day for me). I open the box of ammunition B and note, to my horror, there are only 5 rounds there, I left the rest on the reloading table at home 30 miles away. I shoot the 5 rounds of ammunition B and get a group size of, say, Y. My question is, how do I normalize (or whatever the word should be) Y to be able to get a one for one comparison with the size of the 10-shoot group of ammunition A so I can estimate which is indeed the more accurate loading?

35. I would think that to get a true competence of the difference of A and B and to remove yourself from the equation that you should use a gun vise to hold the rifle instead of just resting it on sand bags. That would remove the variable introduced by both your breathing and trigger finger squeeze, as well as any flinch or jerk you may unknowingly introduce as you fire the piece.

Rx

36. Ooop, not “true competence”, rather true comparance”. (Grammar/spell check changed it on me).

37. 1) since the bad guy is running away while firing back and we can assume that the cop’s ability to hit him decreases with both distance and stress, the actual probability of hitting the bad guy where it matters is best estimated as 1 minus a sequence sum approx 3/10 3/(10 + x1) 3/(10+x2) …

Zeno might hazard a guess, but I don’t know how to get estimates for x without running enough tests to, in all liklihood (!) getting the cop killed far too soon.

2) It is not true that “It is a statistical fact that group size will increase with the number of shots fired”. The limit is set by the range of the weapon and local physical conditions.

Merry Christmas!

38. Sheri says:

revGDright: I know it’s too late now, but I wondered if you had additional ammunition A, why not just shoot a 5 shot group with that?

39. revGDright says:

Yup. That would be the way to do it. Just capture a single 5 shot group with each set. What I’m wondering here though is what happens in this or other cases of process control, etc. when you are trying to compare a phenomena based in disparate sample sizes due to an interruption in the experiment.

40. Sheri says:

Here’s a couple of websites that might help:

http://the-long-family.com/group_size_analysis.htm