Five-all in archery: why it happens, and to whom

Part of the complete guide: How to improve your archery score

Anyone who has shot a few elimination matches knows the feeling: the fifth set closes, you look at the board, and it says 5-5. From that moment two hours of work come down to one arrow each, and most archers experience it as bad luck that happened to them rather than to somebody else.

The calculation says it is not bad luck, and above all says it does not happen at random. Five-all is a frequent outcome, predictable in its frequency, and above all it selects: it does not reach just anybody, it reaches archers who resemble each other. Understanding that changes how you read both the match you have just played and the one you are preparing for, and it concerns the archer before it concerns the rulebook.

The figures that follow come from work I published in the Lab, where the calculation is set out at length and the code to redo it can be downloaded.

Why a match finishes level so often

The reason lies in the way set points are distributed, and in particular in a consequence of that which goes unnoticed: a tied set is not a rare event at all. Even between two archers separated by eighteen points over 72 arrows, which is a gap that is anything but negligible, the individual set finishes level in 23.4% of cases. The stronger archer wins it in 56.3% and loses it in 20.3%.

Nearly one end in four, then, separates nobody, because it distributes one point each. And since five sets distribute ten set points in total, a couple of tied ends are enough for neither archer to reach six, and for the match to finish with five set points each.

Tied sets are frequent precisely because a set is short. Three arrows each is not many, the two partial totals resemble each other often, and the distance between the two archers has little room to show itself. The format is built that way on purpose, to keep the match open, and this is one of its consequences.

How often it actually happens

The frequency can be calculated, and it depends almost entirely on how closely the two contenders resemble each other.

The comparison with cumulative scoring, which is still the format in use for compound, says something clear: the set format produces nearly twice as many ties. Between two identical archers it goes from 13.5% to 20.3%, and at twenty points apart from 5.6% to 10.8%. This is not a hidden defect, it is the direct consequence of a count that wipes out margins at every set boundary, which I have written about in the article on how the set system works.

Across a whole competition

Per-pair figures translate badly into the experience of someone competing, who does not think in probabilities but in days. So it is worth looking at an Olympic bracket of 64 archers, played over 63 matches in six rounds.

In a typical field, with the top around 690 points, the expected number of shoot-offs is about 8.6 across the 63 matches, with a band running from a little under eight to a little over ten depending on how the field is made up. The same bracket played with cumulative scoring would produce 4.7, a little over half as many.

Spread across the rounds, those shoot-offs fall like this: about 3.7 in the first round, 2.3 in the 1/8 round, 1.4 in the quarterfinals, 0.7 in the semifinals. In probability terms that means a little over one match in ten finishes in a shoot-off in the first round, while the gold-medal final goes there between 16.8% and 19.8% of the time, which is almost once in five.

The frequency, then, grows as the bracket advances rather than staying constant, and the reason for that growth is the same one that makes five-all a peculiar situation among all the ones a competition produces.

One last observation about format before getting there. Compound plays its matches differently, adding the fifteen arrows instead of counting sets, and yet it arrives at the same situation: there too, when the two totals coincide, the tie is settled with a single arrow. The frequency changes, roughly halving as we have seen, but the nature of what happens next does not.

Five-all is not a neutral sample

The point that changes everything is that a shoot-off does not occur at random among the matches played, but in proportion to how closely the two contenders resemble each other. Every pair enters the shoot-off statistics with a weight equal to its own probability of finishing five-all, and that probability is highest precisely when the two archers are equivalent.

The effect is what statistics calls selection, and an image makes it clear. If we wanted to judge how reliable a scale is by looking only at the times it showed the same weight for two different objects, we would be looking at exactly the cases in which that scale said least, and we would come away with a harsher view of it than it deserves. Five-all does the same thing: it hands over the matches in which the preceding fifteen arrows failed to decide.

In figures, the mean gap between two archers who actually reach a shoot-off runs between nine and fourteen points over 72 arrows, depending on how the field is made up. An independent check, done on the bracket rather than on the single pair, gives practically the same interval, between 9.2 and 13.7 points.

To give that gap a physical measure: one point on a 72-arrow total is worth about seven tenths of a millimetre of group size, so twelve points of difference, which is the typical case for archers who end up in a shoot-off, is worth a little over eight millimetres of group. These are two archers who look very much alike on the target.

The filter is visible in the ten count

Put that way, selection remains an argument. It becomes something visible if, instead of talking about probabilities, you look at a quantity anyone can count on a scorecard: how many tens the two archers put in across the fifteen arrows of the match.

Take an archer at 688 points against one at 670, so two archers between whom a difference exists and is measurable. Across all the matches they play, the number of tens differs on average by three, and the two shoot the same number in about one case in ten. Look instead only at the matches that ended five-all, and the mean difference falls to one while the coincidence rises to almost one case in three.

They are the same two archers, with the same difference in ability. All that changes is which of their matches you are looking at. Five-all selects precisely the occasions on which they shot almost indistinguishably, and you can read it straight off the count.

For an honest point of comparison it is worth also looking at two genuinely equivalent archers, both at 690 points: between them the unconditioned difference in the number of tens is 2.1 and the coincidence is one case in seven. The filter produced by five-all, in other words, makes two different archers resemble each other more than two genuinely level archers resemble each other in an ordinary match.

It is the most concrete way of saying something that otherwise sounds abstract: having arrived level means, by definition, that the two produced almost identical profiles across fifteen arrows.

The bracket as a second filter

On top of the selection produced by five-all sits a second one, which acts along the bracket. Each round eliminates the weaker archers, so those left in resemble each other more and more, and the pairs that meet become progressively better matched.

The figure is stark: the mean gap between the two contenders goes from 15.2 points in the first round to 4.8 points in the final. Across six rounds it shrinks by two thirds, and the four and a half points of the final are worth about three millimetres of group size.

The two things together explain why the frequency of shoot-offs climbs round by round, from 11.5% in the first round to almost 19% in the final: the further you go, the more the survivors resemble each other, and the likelier it becomes that five sets will not be enough to separate them. The bracket is built to produce balanced finals, and it produces them. Five-all is one of the side effects of that success.

Lower down the levels it gets worse

Everything above holds for a top-level field, but the geometry of the target behaves differently at lower levels, and it behaves in the less convenient direction.

The reason is that wide groups of impacts overlap far more than tight ones. Two archers at 690 points shoot inside groups about four and a half centimetres across; two at 650 inside groups of a good seven centimetres. The same relative difference in ability, between two wide groups, produces impact distributions that are much more alike, and therefore sets that finish level more often and matches that reach five-all more frequently.

For an archer competing at regional level, that means five-all is a situation to count on as ordinary, not an exception to be observed from the height of world finals.

What the calculation does not look at

These numbers come out of a model, and it is worth saying precisely what is left out. No data from real people enters the work: the archers at 690 or 678 are synthetic profiles defined by the width of their group, not anybody's scores. Everything comes from the geometry of the target as fixed by the rules, the rules of the competition formats, and a statistical model of shot dispersion.

The fields of competitors used for the bracket calculations are built deliberately with two declared dials, the mean level and how far apart the competitors are, which is why the bracket results should be read as bands rather than single numbers. When I write that the final goes to a shoot-off between 16.8% and 19.8% of the time, that range is the result, not an approximation of the result.

Finally the calculation assumes every arrow comes out of the same distribution as the others, independently of those before it. We know that is not entirely true, because the tension of a fifth set is not the tension of a first, and the essay declares this assumption its most fragile point.

The only way of not getting there

There is a reading of the same table that interests the archer more than the rule-maker, and it lies in the direction the numbers move. The probability of finishing five-all falls as the gap grows: 20.3% between two equivalent archers, 16.8% at ten points apart, 10.8% at twenty. Every point of real advantage, then, reduces the probability that the match ends up entrusted to a single arrow.

Put in operational terms, the defence against a shoot-off is not built by training the shoot-off. It is built by widening the gap across the fifteen shots that precede it, because it is the gap that decides how often that arrow gets called upon. An archer who gains ten points of level over a season does not only improve their qualification scores: they cut the share of their own matches that finish level by about a quarter.

The same logic explains why the commonest advice you hear on a shooting field, to train the decisive arrow, is only half right. It is worth shooting in practice, because it is the only shot performed in competition under conditions that never arise anywhere else, and arriving at it having never tried it is a pointless disadvantage. But that work does not move the outcome of the shoot-offs you will play, because by then the contest is close to a coin toss by construction.

Anyone coaching can do the sums with their own athlete directly: how many matches did they close in three or four sets last season, and how many did they drag to a fifth? The first figure measures how often the gap showed itself; the second how often the outcome was entrusted to an arrow. They are two numbers you can count off a competition calendar with no instrument at all.

How to read a shoot-off, won or lost

A consequence follows from all this that concerns the Monday after the competition, and it is the reason these numbers are worth knowing.

Losing a shoot-off against an opponent of very close level is an event that happens almost half the time even when the tighter group is yours. That is not a figure of speech: it is the arithmetic consequence of the fact that, between two archers who get there, the deciding arrow comes very close to a coin toss. Anyone who loses while being the more precise of the two has lost a draw that was tipped even slightly in their favour.

The reverse has to be said just as clearly, because it is less pleasant and worth as much. Anyone who wins a shoot-off has legitimately won a match, and nobody takes away the round, the medal, or the result. What they have not won is an argument about their superiority over that opponent.

The right frame, for both, is that a shoot-off does not claim to measure: it claims to close. Its job is to decide who goes through when the competition cannot continue, and it does so in forty seconds, which is exactly what a rulebook is asked for. The misunderstanding starts when closure is mistaken for a verdict, and from there comes a Monday of technical analysis on a shot that contained no technical information.

For anyone coaching the translation is blunt: no training plan should change on the strength of a shoot-off result alone. The fifteen shots that preceded it do contain material worth looking at, because they are fifteen times more informative than the arrow that closed the match.

What to do with it, as an archer and as a coach

The first consequence concerns expectation. If one match in five between equals finishes five-all, and if the share rises through the bracket to almost one in five in the final as well, then the shoot-off arrow is a recurring situation rather than an accident. An archer who shoots a season of elimination competitions goes through it several times, and it is worth arriving having already lived it in practice.

The second concerns how to interpret what happened. Reaching five-all against an opponent is, in itself, information: it says that across fifteen arrows the two of you did not separate, which is to say that you are close. Anyone reading that match as a technical defeat is attributing to their own technique an outcome the contest did not establish.

The third is for coaches. Since five-all selects balanced pairs, and since gaps tighten round by round through the bracket, the work that genuinely moves outcomes in the later rounds has far less to do with the shoot-off arrow and far more with what widens the gap before the fifth set arrives: consistency end by end, which is what allows a match to be closed in three or four sets instead of dragged to a fifth.

Where this goes next

One question this article leaves open on purpose is how much that arrow is actually worth when the moment comes to shoot it. The answer depends precisely on the selection described here, and I have developed it in the article on what the shoot-off arrow decides.

The essay carries the full calculation of the frequencies, the distribution of shoot-offs round by round across twelve different fields, the robustness tests, and the code package with which anyone can redo the sums and check whether the published numbers are the ones the calculation really produces.

Questions I get asked most

How often does an archery match finish five-all?
It depends on how closely the two archers resemble each other. Between two equivalent archers it happens in 20.3% of matches, one in five; at ten points apart 16.8%; at twenty points 10.8%, one in ten. In a bracket of 64 archers the expected shoot-offs run between eight and ten across the 63 matches.

Why do sets finish level so often?
Because a set is short: three arrows each leaves little room for the difference between the two archers to show itself. Even between two archers eighteen points apart over 72 arrows the individual set finishes level in 23.4% of cases, and a tied end distributes one point each without separating anybody.

Does five-all happen more often in finals?
Yes. In the first round a little over one match in ten ends there, while the gold-medal final goes there between 16.8% and 19.8% of the time. The reason is that the bracket eliminates the weaker archers every round, so those left resemble each other more and more: the mean gap between the two contenders goes from 15.2 points in the first round to 4.8 in the final.

Does the set system produce more shoot-offs than cumulative scoring?
Yes, nearly twice as many. Between two identical archers ties go from 13.5% under the cumulative format to 20.3% under the set format, and at twenty points apart from 5.6% to 10.8%. Across a bracket of 64 archers that is about 8.6 expected shoot-offs against 4.7.