Did I really improve?

An archer who changes nothing never shoots the same score twice. A round has only so many arrows and each one lands a little where it happens to land, so the total moves on its own, even when nothing has happened. That movement, measured as the standard deviation of the total, is your floor, and from it comes the threshold below which a difference between two scores tells you nothing, because an archer who had stayed exactly the same would produce the same difference.

The tool answers one precise question. You have two rounds and two different scores: does the gap stand out from the floor, does it sit inside it, or is there not yet enough material to tell? The floor does not come from a table or an archive; it is worked out from your own ring counts. Enter them and press Calculate: below you will find the verdict and the working, step by step, with your numbers. Everything runs in your browser and no data leaves your device.

How it works

Each arrow is a small draw among the rings your shot reaches. From the counts the tool works out two things: the average per arrow, and how far the arrows stray from that average, which is the per-arrow standard deviation. It does not need to know the order the arrows came in or where they landed inside the ring, because the counts are enough for this calculation.

The total of a round is the sum of its arrows. If the arrows are independent of one another, their variation adds up as squares, and the total moves by the per-arrow deviation multiplied by the square root of the number of arrows: that is the floor. In the example round, 654 points from 72 arrows, the deviation is 0.98 points per arrow and the floor comes to 0.98 × √72, about 8.3 points. In other words, two single rounds from that archer, under identical conditions, have to differ by at least 23 points before the gap stands out from chance.

With two rounds the tool brings both totals to 72 arrows, so that rounds of different lengths can be compared too, and combines the two deviations into one, weighted by the number of arrows in each round. From there it works out how much the difference between the two totals moves by chance (its standard error) and builds two intervals around the observed difference, one at 95% and one at 90%.

The decision follows three rules, in this order. If the 95% interval does not contain zero, the gap is a difference. If it does contain zero but the 90% interval lies entirely inside the equivalence margin, the two rounds are worth the same; the margin is half the common floor over 72 arrows, and it never drops below 2 points. Equivalence uses the 90% interval because, to say that two things are the same, two possibilities have to be ruled out (that the second round sits above the margin, and that it sits below it), and each of them is ruled out at 95%. In every other case the verdict is undecided, and the tool estimates how many 72-arrow rounds on each side it would take to settle a gap of that size.

The last block of the result looks ahead: how far apart two rounds have to be, given your floor, for the gap to stand out from chance. For two single rounds the threshold is 1.96 × √2 times the floor, and it falls with the square root of the number of rounds compared on each side.

How to read it

There are three outcomes, not two, and the third is the most common.

Difference. The gap stands out from the floor: with this material it can be told apart from sampling variation. It does not yet say why, whether form, equipment, wind or the day; it says that something is there. It works in both directions, because a drop that stands out from the floor is as much a difference as an improvement.

Equivalence. The gap is so narrow that the two rounds can be called equal within a margin fixed before the comparison. It is a positive result and a rare one, because showing that two things are the same takes a great deal of material: with the arrows of the example round, even with identical averages, it takes more than twenty 72-arrow rounds on each side. To get there, you add up the counts of several rounds in the boxes.

Undecided. It is neither a difference nor a match. The temptation here is to say "nothing has changed", but the data do not support it: the honest answer is "we do not know yet", and the tool tells you how many 72-arrow rounds on each side it would take to find out. That number holds for a gap as large as the one you observed; if the actual gap were smaller, it would take more.

The threshold table is for the comparisons still to come. The threshold falls with the square root of the number of rounds, not with the number itself: halving it takes four times as many rounds. This is why a judgement on a single session cannot be had by paying more attention, only by shooting more sessions. For a coach the practical rule is to check the threshold before commenting on a score: if the gap sits below it, the two scores are the same score seen twice.

The floor and the threshold refer to the number of arrows entered for the first round. If you have added up several rounds in the boxes, the floor is that of the whole block; the comparison between the two rounds, on the other hand, is always scaled to 72 arrows.

After the figures, the block headed "What your numbers say" reads the result back with your own data: which score you went from and to, whether the gap is larger than chance produces on its own, and what the next step is, whether that means adding rounds or moving on to another tool.

Limits

The tool works from ring counts, so it cannot see the shape of the group: that needs the coordinates of every arrow. Two rounds with the same counts can have different groups, one tight and off-centre, the other wide and centred.

The floor worked out here is the sampling floor under fixed conditions, and it is a lower bound. Between one round and the next come the form of the day, the wind, the field and the light, so the variation you see between rounds is always wider than this. For the same reason a difference does not separate a technical change from a different day: two rounds on the same field two weeks apart also differ in the wind, and the floor knows nothing about it. The reverse holds too: if the gap does not even stand out from this floor, there is no need to look for explanations elsewhere.

The calculation treats each arrow as a draw independent of the others, with the same distribution from the start of the round to the end. If the round has a slow warm-up, a drop from fatigue or ends that are very different from one another, the assumption holds less well and the floor should be read as a guide.

The intervals use the normal approximation, which works well on totals of many arrows and less well on short rounds. That is why the minimum is ten arrows, and even with ten the uncertainty on the floor stays large: the tool shows it next to the number rather than hiding it. With 72 arrows it is about 16% either way.

The floor here comes from the counts of your arrows, not from your level. It is the same quantity as in the other tools on the site, the standard deviation of a round's total, but the floor calculator derives it from a model of the target for an archer with your average score: that is why the two figures do not match, and they can differ by a point or more. The model describes a centred, regular group, while your arrows also carry that round's offset and the shots that left the group, and it is the figure worked out from them that describes your rounds.

Finally, the tool compares two blocks of arrows, not a series of competitions over time, and when something has changed it does not say what. It works with any target face scored from 10 to 1: with the indoor triple face, for example, the rings from 5 to 1 simply stay empty.

To follow a series of rounds after a change there is the change detector, which shows the first round at which your level rises above chance. To find out how many rounds you need before and after to confirm a difference there are the floor calculator and the score gap calculator, which also tells you what a gap between two archers in the ranking is worth.