Let us say that you are an excellent archer who can score an average FITA score of 1280 (out of 1440). This is a simulation of what would supposedly happen if you shot 20,000 trials (144 arrows/trial, up to ten points per arrow) at male scoring distances (the distant target is at 90m), assuming I did my calculations correctly (per the model in http://koniaris.com/archery/angles/):

In[112]:=hello[male,1280]

The bottom axis of the plot shows your scores; you can see that they range from below 1240 to above 1320. The height of the dot shows how often you shoot a particular score; you'll note that the highest dots are roughly your average of 1280. It is evident that 1290 will be relatively common, but so will 1270. In fact, most of the time you will be close to 1280 (within eleven points), but every now and then, as you can see, you'll shoot a 1310! (Or a 1255.)

The solid line drawn on top of the dots is the "normal
distribution" that is fit to the simulated data; a normal distribution
has two *parameters,* the *mean* (1280 in this case),
and the width of the distribution, defined by *sigma* (11.2
points in this case). The sigma gives the expected variability from
your average score; as you look below, you'll see that top shooters
are less variable than beginning (say 1000-level) shooters. Most of
the time you will shoot very close to your average, within one
sigma.

However, if you shot a 1300, you might think, "I was a really good shooter today" (i.e., you're improving). However, it could also be the case that you are "merely" a lucky 1280 (average) shooter. Indeed, on two days in a row you might shoot a 1305 and then a 1270, and lament, "oh, I dropped 35 points, that's horrible!" No, it could just be random statistics, and the quality of your shooting could be the same---although that does seem hard to believe, at least at first.

If you are a 1280 shooter, and you're shooting at 70m (for example), I believe that each arrow is rolling an eleven-sided die, where the faces have weights as follows, going X, 10, 9, 8, ... to 0:

In[118]:= N[pvector[findmrad[male,1280]/1000,70] 100,1]

What this means is that you roll an X about 8.2% of the time, a 10 about 18.8% of the time, a 9 about 43.0% of the time, an 8 about 23.0% of the time, a 7 about 6.0% of the time, a 5 about 0.83% of the time, and so forth. So, just as you can roll regular six-sided die and get several "sixes" in a row, you can also "roll your arrow" and get several X's in a row... it's really just luck, I believe. The skill of the shooter sets the relative weights of the faces on the die, i.e., the shooter becomes more likely to roll high scores.

Now, most people who would be a 1280 shooter on average would instead view themselves as 1300 level shooters, because it is human nature to accept the lucky scores as being "correct." This strikes me as a very dangerous idea, because when an unlucky score will show up, the shooter will be very discouraged---despite shooting at a consistent level of quality! It seems hard to believe that a uniform standard of shooting could produce two scores that differ by (say) forty points, but this statistical distribution is actually quite wide with regard to the span of the curve, and big differences in score are actually to be expected.

The most "dangerous" thinking that I am aware of in archery is that if an arrow hits the X, it's a good shot, and if not, it's a bad one. This is absolutely absurd and contrary to the nature of statistics. The correct concern for any student of archery to slowly and steadily improve his or her average, and not be excessively concerned with statistical fluctuations, because they do not necessarily mean much.

The rest of this document shows the expected distribution for male and female shooters around various average scores. This simulation does not degrade the shooter in any way, i.e., it will not account for being tired, sick, etc.; it's just a simple simulation predicting scores that I hope to have implemented correctly! As always, if you should note a problem, please be so kind as to email me so that I can fix it. (I tried to figure this out on paper but then decided that I felt like writing a Monte-Carlo program instead---but the results seem so simple that I'll have to return to paper.)

There is a related computation that's particularly interesting: I want to look at the "Olympic Elimination Rounds," where two archers go "head-to-head," and try to see why there are so many upsets. While everybody claims the upsets are the results of "the mental game," and that might be true, I suspect that the luck plays a huge factor given that only 18 or 12 arrows are used, so the distributions become even wider than when 144 arrows are used. (More later....)

Males shoot at 30m, 50m, 70m, and 90m.

In[106]:=hello[male, {900, 950, 1000, 1050, 1100, 1150, 1200, 1250, 1300, 1320, 1340, 1360, 1380, 1400}];

Females shoot at 30m, 50m, 60m, and 70m.

In[107]:= hello[female, {900, 950, 1000, 1050, 1100, 1150, 1200, 1250, 1300, 1320, 1340, 1360, 1380, 1400}];