Matteo Campagna
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How many points it takes to show you have improved

How much a gap in the ranking is worth, and how many rounds it takes before and after a technical change to say that it worked.

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A round's total holds together skill and that day's luck, and the share that belongs to luck is bigger than people think: two identical top-level archers, over 72 arrows, end up five points apart on average. This calculator uses the same model as the essay on the set system and the article on how many points it takes to say an archer is better.

Example with illustrative values, at 650 points over 72 arrows. Top, one round: the bell curves of the totals of two archers 5 points apart, with the 95% threshold at 16.1 points; the probability that the archer ahead is the more precise is 70%. Bottom, the average of 11 rounds before and 11 after an improvement of 5 points: the 95% threshold drops to 4.8 points, and the improvement clears it. One round: the totals of two archers example 630 640 650 660 670 95% threshold 5 points behind ahead Average of 11 rounds, before and after 630 640 650 660 670 95% threshold 5 points before after points over 72 arrows
Example with illustrative values. Two archers, one averaging 650 and one averaging 655 points over 72 arrows, with the spread the model gives at 650 (a standard deviation of 6.9 points). In a single round their bell curves overlap for the most part, and the 650 archer finishes ahead about 3 times in 10: that is why a 5-point gap is worth a 70% probability that the archer ahead is the more precise, the calculator's first result. With the average of 11 rounds before a change and 11 after it, the curves narrow and a 5-point improvement clears the 95% threshold: that is the second result. In each half the thin line marks that threshold, the distance beyond which a difference counts with 95% confidence.

Enter your numbers and press Calculate: below you will find the two results, a chart of your numbers drawn like the figure above, a comment that reads them against the numbers you entered, and the working, step by step.

Your numbers

The model's limits

The model looks only at arrow-to-arrow chance, with the archer's form constant from the first arrow to the last. In real life sleep, the day, the mind and the wind come on top: nearly all of them widen the range. That is why the values are an optimistic estimate: gaps say a little less, and the rounds needed are a few more. The confidence about the order assumes nothing is known about the two archers beforehand; in a real ranking, neighbours are almost always close in level too, and a gap says even less.

The model is the pure bell curve of the essay on the set system, and the number it starts from has a name: the standard deviation of the 72-arrow total is the floor. At 650 points this calculator gives 6.9, while the floor calculator, which uses the table in the essay Il pavimento (The floor), gives about 7.0 with the pure bell curve and 7.5 with the model with tails, the one that allows for the few arrows that leave the group. The difference comes from that and from the way the table is read between levels: this calculator stays calibrated on the published tables, the other covers a wider range of levels.

What to make of it: do not read too much into one round against the next, and keep an average of your last four rounds, which has half the uncertainty of a single one. To compare two rounds from their ring counts there is Did I really improve?; to follow your rounds one at a time after a technical change there is the change detector.