How much your score moves when nothing has changed

A 72-arrow total changes from one round to the next even when nothing in your shooting has changed, because the target cuts the cloud of arrows with hard edges. This calculator answers three questions: how much your total moves by pure chance, how large a difference you need between two blocks of rounds to say that your level has changed, and how many rounds it takes to confirm a difference you care about. It uses the definitions and the numbers of the essay Il pavimento (The floor, edition 1.0). Enter your level and, if you have them, your recent round totals, then press Calculate: below you will find the results and the working, step by step, with your numbers.

How it works

The floor is the standard deviation of a round's total when the archer's state stays put: same form, same equipment, same conditions. The arrows still form a cloud, and the target divides it with hard edges: an arrow just on one side of the line scores 9, a few millimetres further over it scores 10. So even when the shot stays identical, the number of arrows landing in each ring changes from one round to the next, and the 72-arrow total carries that movement with it.

The essay works it out without any competition data, from the geometry of the 122 cm face. From the level it gets the average per arrow (630 points divided by 72 arrows is 8.75); from that, the width of the cloud that produces it; from the cloud, the probability of landing in each ring; and finally the standard deviation of a single arrow, which multiplied by the square root of 72 gives the floor. At 630 points a single arrow has a standard deviation of about 0.96 points, and 0.96 × 8.49 comes to about 8.1.

The calculation is done in two scenarios, and the two values should be read as a pair. In the first, the pure bell curve, every arrow comes from the same cloud. In the second, the model with tails, a small share of arrows falls outside the normal group (just over two in a hundred at 630 points) and weighs on the total more than its number suggests, because an arrow outside the group can cost eight or nine points, while a perfect arrow gains little more than one. The calculator uses the essay's table, shown below, and reads values between levels off a straight line. When a single number is needed it takes the one with tails, because the essay always picks the end that makes a change harder to declare.

Floor in points, recurve at 70 m on the 122 cm face. ° Above 665 points the essay holds the share of arrows outside the group constant rather than extending it.

Three formulas follow from the floor, and they are what the calculator works with. The first concerns two single rounds: their difference adds up the chance in both, so it moves by √2 times the floor, about 1.4 times, and two scores closer than that are not even unusual. To say at 95% that two rounds come from different levels, the difference has to exceed 1.96 × √2, about 2.77 floors: at 630 points that is 25 whole points. The second concerns two blocks of rounds, because the average of several rounds moves less than a single round: the threshold becomes q × s × √(1/n₁ + 1/n₂), where n₁ and n₂ are the rounds in the two blocks. The third turns the question round and gives the number of rounds needed to confirm a difference Δ with 95% confidence and an 80% chance of seeing it when it is there: 2 × (1.96 + 0.84)² × s² / Δ², rounded up. At 630 points, for 5 points, that is 49 rounds per block.

If you enter your round totals, the calculator works out their standard deviation, with the n − 1 divisor the essay asks for, and uses it in place of the floor. For comparing rounds it is the right number: the floor is only the share of chance that comes from the arrows, while your rounds also carry form on the day and conditions on the field, and they move more. A standard deviation taken from a few rounds is uncertain too, though, so in the threshold the 1.96 gives way to the Student t quantile, with as many degrees of freedom as there are rounds minus one: with six rounds it is 2.57. If your standard deviation comes out below the floor, which can happen with few rounds, the calculator uses the floor, which within the model is the minimum.

How to read it

The floor is not a mistake you make; it is the margin within which your score moves on its own. A 630 archer who shoots 633 on Saturday and 624 on Sunday has dropped nine points, and it is natural to look for the cause in the shot; but nine points sit inside the typical gap between two rounds, which at 630 is about 12 points, and a long way from the 25 or so that two single rounds need to clear the 95% threshold. With those two results a decline may or may not be there: it simply cannot be told apart.

Read the threshold with the same care in both directions. Clearing it does not prove that your level has changed: it says that, had it stayed the same, a difference this large would be rare. Staying below it does not prove that nothing has changed: it says that these rounds do not show it yet. When the averages you entered stay below the threshold, the honest answer is to wait for more rounds, not to adjust your technique on the strength of that difference.

The rounds needed are a planning number, to be looked at before you collect the data, not after. If they seem a lot, that is not a fault in the calculator: halving the difference you want to see quadruples the rounds it takes. This is why, when the question about competition scores becomes out of reach, it pays to measure further upstream, on ends and on where the arrows land, where there are many more observations.

Under the results, in the box headed "What your numbers say", the calculator does this reading with your own data: the range your rounds cover, whether they move more than the floor, whether the difference between the blocks clears the threshold, and what the next step is.

Finally, the level to enter is your expected score, the average of many rounds, not today's result. If the standard deviation of your rounds is much higher than the floor, that gap is useful information: it is the share of movement that comes from form on the day and conditions on the field, and it is the part an archer and a coach can work on. If what you want to know is how much a gap between two archers in the ranking is worth, there is the score gap calculator. To compare two rounds from their ring counts there is Did I really improve?, and to follow your rounds one at a time after a change there is the change detector.

Limits

The model holds for recurve at 70 metres, on the 122 cm face, with 72-arrow rounds and a centred cloud. For compound and barebow the essay proposes that the curves are different, but they have to be measured, and the calculator does not make them up. Outside the 520 to 700 point range I use the value at the nearest edge, and say so. Between two rows of the table the value is read off a straight line, whereas the essay calculates it directly: the difference is a few hundredths of a point, and at 670 points it is enough to move the threshold between two rounds by one point (18 here, 17 in the essay).

The pure bell curve column is the same model as the essay on the set system, which is published: at 690 points both give 4.6. The score gap calculator uses the pure bell curve alone, calibrated on the published tables of that essay, which is why at 650 points it gives 6.9, where the pure bell curve read off a straight line here gives about 7.0 and the model with tails 7.5. The floor is the same: what changes is the scenario and the way the table is read between rows.

The floor is a minimum, not a forecast. Real rounds always move more, because form on the day and conditions on the field come on top of the chance in the arrows: in the essay's three-layer scenario, at 630 points, the floor is about 8.8 points out of an overall movement between rounds of about 20. So if you do not enter your rounds, the threshold and the rounds needed are the lowest possible values, and with real rounds they grow.

The calculation treats arrows as independent, and this is its most delicate point. If arrows tend to drift together, even very slightly, the floor rises quickly: with an average correlation of one hundredth between pairs of arrows, at 630 points it goes from 8.8 to about 11.5.

The standard deviation of your rounds, when you enter only a few, is very uncertain: with 6 rounds the 95% interval runs from 0.62 to 2.45 times the value obtained, with 10 from 0.69 to 1.83, with 20 from 0.76 to 1.46. The Student t widens the threshold to allow for this, but it cannot make an estimate from three rounds precise. The rounds also need to be comparable, with the same distance and format: mixing 18 and 70 metres breaks the calculation.

The totals you enter are used to estimate the standard deviation, and the calculator treats them as a separate window, with the number of rounds minus one as degrees of freedom. If they are the very rounds of the two blocks you are comparing, the essay does the calculation differently: it measures the standard deviation within each block and uses n₁ + n₂ − 2 degrees of freedom, so that with three rounds a side q is 2.78. The formulas hold for two independent blocks of rounds. If the two blocks are paired, or if the rounds come in clusters (two on the same weekend, for example), the essay corrects the calculation, and the calculator does not. The number of rounds needed uses the normal distribution: the exact calculation, which allows for the standard deviation being estimated too, adds one round, and anyone planning a comparison would do well to add one or two.

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