How big your group is
From the ring counts of a round, without marking where each arrow landed, the size of your group in millimetres and how it compares with your level.
A round's score tells you how many points you made, not how big your group is. That information is there all the same, inside the ring counts: a tight group fills the 10 and the 9, a wide one pushes arrows out towards the 8 and the 7. This calculator starts from how many arrows landed in each scoring ring and works out the size of your group in millimetres, with no need to mark where each arrow went. Choose the target face, enter the counts from a round (a full 72-arrow round is ideal) and press Calculate: below you will find the results and the working, step by step, with your own numbers. Everything is calculated in your browser and no data leaves your device.
How it works
On the target a group of arrows looks like a cloud, dense in the middle and thinner towards the edge. The calculator describes it with a single number, σ, the standard deviation of the arrows horizontally and vertically, measured in millimetres. Once you know σ and the geometry of the face, meaning the radii of the rings, the share of arrows expected in each ring follows from a known formula. The calculation can therefore be run backwards: start from the counts you observed and look for the σ that makes them most likely. This is maximum likelihood estimation, and the calculator does it by trial, one millimetre at a time.
The method has been tested on 78 arrows from one archer whose coordinates were also known: from the rings alone it gave 66 mm, against 66.6 mm for the spread measured on the positions around the centre of the target, a difference of 1%. It is a test on a single archer, useful to see that the method works, not a validation across many archers.
From the same counts the calculator also estimates the share of arrows outside the group, with a two-part model: the core, which is your normal group, and a much wider component that collects the arrows that went off badly. The weight of the wide component that best explains your counts is ε. The other numbers are direct calculations: the 90% diameter comes from σ, the X rate from the counts and the floor from the spread of the single-arrow scores.
How to read it
Under the results, in "What your numbers say", the calculator comments on your data in a few sentences: where your arrows landed, what the σ they produce means, how this round compares with your usual score and what to do next.
The spread σ is the central number, because it measures the technical size of your shooting: the smaller it is, the tighter the group. For reference, in the norm the calculator uses for recurve at 70 m an archer who shoots 600 has σ of around 101 mm, one who shoots 640 around 74 mm and one who shoots 680 around 50 mm. Where that norm comes from is explained under the limits, at the bottom of the page.
The 90% diameter turns σ into something you can see on the boss: nine arrows in ten sit inside a circle of that diameter. When coaching it is the easiest measure to use, because "your group today is 24 cm" makes sense straight away.
The share outside the group counts the arrows that do not belong to your normal group: messy releases, gusts of wind, arrows that went off badly. The essay Il pavimento (The floor) proposes, as a scenario still to be tested and not as a measurement, that this share runs from about 6% of arrows for a 440 archer to about 2% between 630 and 665 points. The further your share sits above those values, the more the margin to recover lies in the arrows that left the group rather than in the size of the group. Below 300 arrows, though, read this number as an indication.
The X rate is the share of arrows inside the X ring. It measures fine aiming, counts in tie-breaks and rises quickly as the group tightens.
The floor over 72 arrows is the standard deviation that the total of that round carries by pure chance, the same quantity the other tools on the site use. Two rounds are not compared against the floor, though: their difference adds up the chance in both, and the typical gap between two single rounds is about 1.4 times the floor, or √2 times. If two rounds differ by less than that there is nothing to read into it, and no reason to change your technique over a result like that. For a proper comparison of two rounds there is Did I really improve?, and to find out how many rounds it takes to show an improvement, the floor calculator.
The comparison with the norm puts your σ next to the typical value for recurve archers of your level at 70 m. Within 8% either way the group is in line with the level (the 8% band is a convention of the calculator, not a measurement); tighter means you group better than the typical archer of your level, wider means the margin to recover lies in your grouping.
Limits
The calculator assumes the group is centred on the middle of the target. If the group sat off centre that day, because of a sight that needed adjusting or because of the wind, the offset ends up inside σ: the value you read is a ceiling on your technical spread, which may be a little smaller. Separating the size of the group from its position takes the coordinates of the arrows, in other words plotting: you can mark them with the tap target or on paper with the plotting sheet.
Rings are circles, so they say nothing about direction: a group stretched vertically and one stretched horizontally can give the same counts. σ is an average spread across the two axes, and finding out which one dominates again takes the position of the arrows.
The estimates depend on the number of arrows. With 12 arrows the calculator already gives a number, but a full 72-arrow round is ideal; the share outside the group, which by definition rests on few arrows, only becomes reliable from about 300 arrows upwards, which means adding several rounds together.
The comparison norm exists for recurve at 70 m on the 122 cm face. Its values were derived with this same method from the ring counts in an archive of competition results analysed for an earlier project, which is not published: read them as a rough reference, not as a norm anyone can check. On other faces σ is worked out in the same way, but there is no reference band. The model also assumes that every face is a full ten-ring face: this is an assumption of the calculation, and on reduced faces, such as the six-ring compound face or the indoor triple faces, it does not hold, because an arrow off the face is treated as if it had landed beyond the 1 of a full face, and the estimate of the arrows outside the group suffers for it.
Finally, the floor over 72 arrows only covers arrow-to-arrow chance within that round. From one day to the next, sleep, the mind, the wind and the conditions on the field come on top and widen the range: what you see here is an underestimate of how far two rounds shot on different days can move apart. It comes from your counts, not from your level, which is why it can differ by a point or more from the value the floor calculator gives for your average score, which comes from a model of the target with a centred, regular group.