Archery scores at different distances: Park 2014 study

Park proposes a scoring model in which the spread of arrows on the target has two parts: one that grows in proportion to distance and one that grows with its square. He uses it to estimate how much of the lost score comes from each kind of error. The model starts from the one Archery Australia has used since the 1970s, in which the archer's level is an index (100 was the men's world record of the time; today compound records correspond to about 125 to 130 and recurve records to 115 to 120), and adjusts it so that, without wind, an archer scores the same at 70 metres on a 122 cm face and at 50 metres on an 80 cm face, as competition results suggest.

Checked against men's qualification scores from five low-wind events (the 2007 and 2009 World Championships and three 2009 World Cup stages) at all four distances, the new model has a root mean square error of 0.54 points per distance for compound, against 0.89 for the Archery Australia model, and 0.99 for recurve, against 0.80. It follows that in light wind, at 70 metres, 75.9% of lost points come from errors that grow with distance and 24.1% from errors that grow with its square; at 50 metres the shares are 81.5% and 18.5%.

In the author's classification, the only errors of the second kind are wind drift and bow cant that varies from one arrow to the next. With no wind, then, that quarter of the points lost at 70 metres would all come from bow cant. For a recurve archer with an index of 105, who would average 336.7 over 36 arrows at 70 metres, that means about 5.6 of the 23.3 points lost, corresponding to cant varying by plus or minus 0.55 degrees. At the 2013 World Championships the afternoon wind, around 3 to 5 m/s, cost recurve archers about 5 points over 36 arrows at 50 metres; in the example of the eighth-placed archer, of 30 points lost at 50 metres about 5 come from wind, 5 from bow cant, 3 from sight setting and about 17 from everything else.

The arrow group is described by a circular normal distribution: the radial distance from the group centre depends on a standard deviation calculated from the archer's index and the distance, and the expected score of each arrow comes from laying this distribution over the target rings, with an arrow diameter of 5.0 mm outdoors. That groups are roughly normal had been checked by photographing every archer's groups at 50 metres at a 2013 World Cup stage. The split between the linear and quadratic parts (0.815 and 0.185 at 50 metres) is fixed by requiring the same score at 70 and 50 metres for an archer with an index of 100.

For the comparison with competitions, Park took the distance scores from 144-arrow qualification rounds (men at 90, 70, 50 and 30 metres), choosing low-wind events among those he had attended, and summarised them with regression lines against ranking position, leaving out the tail; at 30 metres, where the best archers shoot perfect scores, he used a hyperbolic curve. The effect of bow cant was estimated with a Monte Carlo simulation of 28,800 shots, moving arrows sideways by varying amounts until the predicted points were lost; the conversion to degrees uses a 213 newton recurve with 450-spine X10 arrows of 23.7 grams at 61.4 m/s.

The 24.1% share rests on two of the author's choices: that scores at 70 and 50 metres are equal, fixed at an arbitrary index of 100, and that the only errors growing with the square of distance are wind and bow cant. Park himself notes that the finger release on a recurve might have a quadratic part, in which case the points attributed to cant would shrink. The model adds the two parts directly within the standard deviation and then reads their proportion as the proportion of lost points: this is a simplification, not a measurement of any archer's errors.

Wind conditions at the events are not recorded: the "low-wind" events are chosen from what the author had seen on site, and he acknowledges that wind varies across the days of a competition. The comparison concerns ranking averages, not individual archers, and for recurve the new model is no more accurate than the old one (0.99 against 0.80 points). The 0.55 degree figure is worked backwards from the model, not measured on archers, and the 3 points for sight setting also come from another paper by the same author.

The useful idea is the distinction between errors that grow with distance (aiming, release, draw length, anchor, differences in mass and spine between arrows, a badly set sight) and errors that grow with its square, namely wind and a bow canted differently from one arrow to the next. For the latter, moving from 50 to 70 metres costs far more, and cant matters more on a recurve, which launches the arrow more slowly and cannot carry the bubble level allowed on a compound. According to the model, cant varying by just over half a degree either way would cost a top-20 world archer more than 5 points over 36 arrows at 70 metres; and since in qualification a point is worth about one ranking place, that is not trivial.

These are model estimates, not measurements on archers, and they give an order of magnitude rather than telling any individual archer how much they lose. The author adds a hypothesis still to be tested: archers who do better indoors at 18 metres than their outdoor scores would suggest might have a larger share of quadratic errors, perhaps in bow cant itself.