Matteo Campagna
ITEN
Menu
Menu

The round at which an improvement really shows

Enter your level before the change, how much your total usually fluctuates and the scores of the rounds that came after: the detector follows them one by one and shows the first round at which the change can no longer be put down to chance.

Last updated:

After a technical change, a new training programme or new equipment, the question is always the same: has your score gone up because you have improved, or because your last few rounds happened to go well? The detector answers it by following your rounds one at a time, in the order you shot them, and shows the first one at which the change can no longer be explained by the normal fluctuation of your total. Enter your numbers and press Calculate: below you will find the result, two charts, a round-by-round table and the working, step by step. The calculation runs in your browser and your scores never leave your device.

Your numbers

The average of your rounds before the change.

How much your total usually fluctuates from one round to the next (the standard deviation of your totals). For a recurve archer on 630 over 72 arrows the essay Il pavimento (The floor) gives between 8.1 and 8.8 points; the floor calculator works it out at your level or from your totals.

The size of the improvement you want to see, for example 5 or 6 points.

Separated by commas, spaces or new lines. At least three rounds are needed.

How it works

The method is called CUSUM, short for cumulative sum. It comes from industrial quality control, where it is used to notice as early as possible that a machine has started producing parts out of tolerance, and it works the same way with an archer's scores: rather than looking at one round at a time, it keeps a running count of how far the rounds are drifting from the starting level, all of them together.

At each round the detector updates two sums. The first looks for an improvement: it adds how much the score exceeds the starting level plus a small tolerance, which is half the change you want to detect. With a baseline of 625 and a Δ of 6, the tolerance is 3 points, so a round of 631 adds 3 points to the sum and a round of 627 takes 1 away. The second sum does the same in the other direction and looks for a decline. Neither can go below zero: when the rounds come back around the baseline, the sum empties and starts again.

If your level has not changed, the scores go up and down around the baseline, the tolerance deflates them and both sums stay close to zero. If your level has moved, one of the two keeps growing round after round and sooner or later crosses the threshold, which is five times the floor: with a floor of 9 points, 45. The round at which that happens is the round of the detection. With the values in the example, a floor of 9 and a Δ of 6, an archer who has not changed at all sets off a spurious alarm about once every 125 rounds: the figure comes from a simulation of 20,000 series of rounds, with totals normally distributed around the baseline.

Illustrative example of the detector, with made-up rounds Top, 14 made-up scores around a baseline of 625: up to round 8 the level in the example stays at 625, from round 9 it rises to 634. Bottom, the upward sum: before the change it climbs to 11 at round 4 and falls back to zero, for the last time at round 8; then it grows round after round and at round 14 it reaches 48, above the threshold of 45 points: that is the round of the detection. Round scores Upward sum 620 630 640 650 0 20 40 from round 9 the level rises to 634 baseline 625 threshold 45 = 5 × floor rises, then empties last zero: round 8 detection: round 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 round Illustrative example of the detector, with made-up rounds Top, 14 made-up scores around a baseline of 625: up to round 8 the level in the example stays at 625, from round 9 it rises to 634. Bottom, the upward sum: before the change it climbs to 11 at round 4 and falls back to zero, for the last time at round 8; then it grows round after round and at round 14 it reaches 48, above the threshold of 45 points: that is the round of the detection. Round scores Upward sum 620 630 640 650 0 20 40 from round 9: 634 baseline 625 threshold 45 rises, then empties last zero: round 8 detection: round 14 1 3 5 7 9 11 14 round
The detector on an example. Made-up rounds, with a baseline of 625, a floor of 9 and a Δ of 6: up to round 8 the level stays at 625, from round 9 (grey area) it rises to 634. Taken one at a time, the rounds cannot show it: round 12 drops to 621, below the baseline. The sum, instead, adds how far each round goes above 628, the baseline plus the 3-point tolerance, and takes off what falls short: at round 14 it crosses the threshold of 45. The last round at which it stood at zero, round 8, shows where to look for the start of the change.

The advantage over comparing two blocks of rounds is that you do not have to decide in advance how many rounds to shoot: the detector stops as soon as the rounds are enough. A clear jump is picked up in a few rounds and a small change takes more: in the same simulation, again with the values in the example, a real improvement of 12 points is picked up after about 6 rounds on average, one of 6 after about 14 and one of 3 after about 38.

How to read it

The result can be one of three kinds. If it shows an improvement detected at round 10, from that round on your level has risen by more than the normal fluctuation of your total can explain. A decline detected reads the same way in the opposite direction, and it makes sense to find out what has changed: equipment, training load, technique. If nothing is detected, everything is still within the noise of the floor. Just below the result, the panel headed "What your numbers say" puts it in terms of your own data: how far you have moved from the baseline, what that means given your floor and what to do next.

The round of the detection is not the round at which the change happened: it is the one at which it became provable. The change usually began a few rounds earlier, and the best clue to where is the last round at which the sum stood at zero: from there on it stayed above zero until it reached the threshold. The working below the result names that round with your numbers, and you can see the same thing in the second chart.

Silence from the tool is information too. After a few rounds it only means that the change cannot be shown yet, and the thing to do is keep collecting scores; after many rounds, if the sums stay low, the improvement you feel is probably not there, or it is smaller than the one you are looking for.

In the table each row is a round, with the two sums straight after that score: it is the most direct way to see the mechanism at work and to tell which rounds pushed the sum towards the threshold. After a detection, if you want to keep following the new level, start again with a new baseline, the average of the rounds after the change.

Limits

The first limit is the baseline. It has to be the average of a block of rounds shot before the change: if you include rounds that had already improved, the baseline goes up and the detection comes late or not at all. The rounds also have to be comparable, with the same distance, format and number of arrows, because mixing 18 m and 70 m scores breaks the baseline.

The second is the floor, which sets the threshold. Underestimate it and the threshold drops, so the alarm goes off too easily; overestimate it and the detection comes late. It is better to work it out from your own rounds, the same ones that give you the baseline, than to use a typical value. Here the floor is the standard deviation of your totals, the same quantity as in the other tools on the site, and it should not be confused with the typical gap between two single rounds, which is about 1.4 times as large (√2).

The rate of spurious alarms, about one every 125 rounds, also holds only for values similar to those in the example, with a Δ of around two thirds of the floor. With a smaller Δ the tool picks up small changes sooner, but raises a spurious alarm far more often: with a floor of 9 and a Δ of 3, in the same simulation, about once every 43 rounds. With a larger Δ spurious alarms become rarer (with a Δ of 9, about one every 460 rounds), but the tool only picks up clear jumps.

A gradual change, growing a little at every round, is picked up later than a step change, because the sum takes longer to build up. In the same way a single round far from the others carries a lot of weight: a total that collapsed because of the wind or an equipment problem can be enough on its own to trigger a decline. Before reading the result, check whether the series has a score out of scale and ask yourself whether that round was really comparable with the others.

Finally, the detector looks only at totals: it tells you whether and when your level has moved, not why. For that you need the group, the technique and what happened between one round and the next.

If you already have two blocks of rounds and want to know whether the difference between their averages is bigger than chance, or how many rounds you need before and after, there are the floor calculator and the score gap calculator; to compare two single rounds from their ring counts there is Did I really improve?.