Arrow groups at world archery events: Park 2014 study

The authors digitised the arrow holes one by one on the target faces of two major 2013 events: about 20,000 at the first World Cup stage in Shanghai and about 10,000 at the World Archery Championships in Belek, Turkey. Three things come out of those data.

The first is what a point is worth. Excluding the tail of the field, where archers are simply not competitive at that level, each ranking place was worth about 1.4 points in recurve and 0.7 in compound, in both cases about 0.1% of the maximum score per place. And the ranking place decides the outcome: bar the occasional upset, the medals went only to archers placed in roughly the top twenty. To sit around twentieth in Shanghai called for 1320 points in men's recurve and 1325 in women's recurve over 144 arrows, and 695 in men's compound and 675 in women's compound over 72 arrows.

The second concerns distance. The men's recurve scores at 70 metres on the 122 cm face were very similar to those at 50 metres on the 80 cm face, and the same held for the women. If group size grew in proportion to distance, a score at 50 metres on the small face would correspond to one at 76 metres on the large one: group size therefore grows faster than in proportion. The gap between the two distances, small as it was, widened as the archer's ability increased.

The third is the most striking. Moving each digitised group to the position that would have given the highest score, many archers would have gained points: about 3 per archer on average over 36 arrows at 50 metres, with far worse individual cases, and the loss was not confined to the weaker competitors. The one exception is men's compound, where the top 22 in the ranking gave away six points between them, three of which belonged to a single archer. On group shape, the ratio of horizontal to vertical spread averaged 1.15 in Shanghai, where the wind was below 1 m/s, and 1.25 in Belek, where it blew harder: groups are wider than they are tall, and this happens even in light wind.

The two events were chosen for contrast. The Shanghai World Cup was held in a well-sheltered stadium and the wind during the ranking round and most of the matches was estimated at under 1 metre per second; the World Archery Championships in Belek were close to the sea, with a quartering wind from the rear right of up to 5 metres per second in the afternoons and gusts measured at over 20 metres per second during the head-to-head matches. 2013 was also the last year in which recurve archers at international events shot on individual target faces at the shorter distances, which is what made this data collection possible.

In recurve the ranking round was 144 arrows: 36 at each of 90, 70, 50 and 30 metres for the men and at 70, 60, 50 and 30 for the women, with the 1220 mm face at the two longer distances and the 800 mm face at the two shorter. Compound archers shot 72 arrows at 50 metres on the 800 mm face, in two sets of 36. Archers could take practice shots only at their longest distance.

In recurve each individual face was photographed after the 50 metre distance, and in Shanghai replaced and photographed again after the 30. For compound in Shanghai, the faces were photographed after the practice shots (typically 12 to 30 arrows) and after the first set of 36 at 50 metres, then some were replaced and all were photographed after the second set. Every group was digitised and analysed, with the digitisation cross-checked against the totals the archers had recorded for those arrows; the analysis was only partly automated, because overlapping holes and arrows close to the scoring lines needed manual assessment.

For the score loss due to group position, the coordinates of each group's centre were computed and the group moved to the centre of the face; the score in the new position was recalculated allowing for arrow diameter (5.0 mm, since an arrow touching a line takes the higher score) and, where it came out higher, the difference was recorded as the potential score loss. In a small number of cases the group was moved slightly off-centre to obtain a higher score. Group shape was described by the ratio of horizontal to vertical standard deviation, and group tilt by the correlation between the lateral and vertical position of the arrows. The expected score estimates use a model long in use at Archery Australia, in which group spread grows with distance according to an exponential law with three fixed coefficients.

The study observes rather than intervenes: it is a snapshot of two 2013 events under the format of the time, and that format changed the following year. There is no comparison with any different condition and no archer was followed over time.

The score loss calculation has a limitation the authors state themselves, and it runs towards understating the effect: over the 36 arrows the archers may have moved their sights, which was not recorded because following it would have been impractical and intrusive. The group seen on the target is therefore wider than the archer's own spread with a fixed sight, and the estimated gain from moving the group to the centre is smaller than the real one. It also remains a calculation made after the fact: it says how much could have been gained by already knowing where all 36 arrows went, which is information an archer does not have while competing. The authors acknowledge this, and write that estimating the centre of a group from very few data points, in the middle of a competition, remains an open problem.

Weather conditions at past events are not recorded by World Archery, so the comparison between Shanghai and Belek rests on wind estimates made on site. The shape of the groups, wider than tall even in light wind, is left unexplained: the authors note only that in that wind it cannot be the wind.

Some of the most interesting readings are interpretations rather than measurements. That the difference between compound and recurve in the strong-wind matches comes down to how compound archers operate their release aids is the author's observation of what he saw on the field. The law by which group size grows with distance comes from a pre-existing Archery Australia model, with coefficients chosen years earlier so that the world record of the 1970s corresponded to an index of 100: a useful convention, not a result of this work. Finally, the improvement of the Australian archers between the two events, from 17 points lost to 3, is a comparison of six people in two different settings with no control group.

This is the study that puts a price on a point. In the ranking round of an international event a point is worth about one ranking place in recurve and one and a half in compound, and the ranking place decides almost everything, because medals almost always go to archers inside the top twenty. From there follows the part that matters to an archer: three points left on the target because the group was not centred are worth three places, and three points over 36 arrows is what the archers at those two events left on average.

The observation holds at the top as well. Apart from men's compound, where the first twenty-two had their groups essentially centred, the loss was not a problem of the weaker field: it showed up in the groups of leading archers too, one of whom would have scored 353 instead of 347 at 30 metres and another 341 instead of 335 at 50. And the calculation is a cautious one, because the sight may have been moved during the 36 arrows: the real margin is probably wider.

On distance, the useful result is that scores at 70 metres on the large face and at 50 on the small one resemble each other, whereas in proportion the ones that should match are 76 and 50. Group size therefore grows faster than distance, and the gap between the two widens as ability increases: the authors infer that the factors degrading a score faster than linearly with distance, such as wind drift or variation in the bow's lateral cant, weigh more and more heavily as an archer improves.

One last thing about group shape. On average it is wider than it is tall, 1.15 in Shanghai and 1.25 in Belek: lateral spread is the bigger problem, even when there is no wind. In recurve the groups also had a slight tilt from upper right to lower left, with a mean correlation of 0.1, in a field of almost entirely right-handed archers; in compound that tilt was absent.

None of this is a procedure: the authors describe what happened on those target faces and say that the best way for an archer to adjust the sight during competition remains a matter of ongoing research.