The number 7 is number that gives you the shortest path to a valid evaluation
IF you are evaluating group sizes (the distance between the two furthest spaced holes (extreme spread) in the group). It is explained for evaluating groups in
this paper by Geoffrey Kolbe that for 15% error tolerance, 42 shots in 6 sets of 7 will get you there. For 5% error tolerance, 385 total shots (55 groups of 7) have to be fired. The same would apply to analyzing velocity by the ES numbers of your strings alone. Kolbe points out 5-shot samples are almost as good as 7-shot samples (45 rounds total, or 9 groups of 5, verses 42 rounds total in 6 groups of 7), though you do use up more target paper.
If you know the location of each shot, rather than firing 42 shots, it takes about fifteen shots to get 15% tolerated error based on radial SD. Kolbe sites a monograph with proof that radial SD is the best measure to use. If you plot radial SD and CEP68%, they are very close for any significant group shot size, so I doubt most shooters will find there's any practical advantage to using one over the other, though CEP50% is more commonly used for CEP.
One problem you run into with the sample standard deviation formula used in every chronograph I've seen is it suffers from what is called square root bias when the sample size gets small. For sample sizes of 7 or smaller, it turns out you can get a better estimate of population standard deviation simply by multiplying the ES by a statistic called Xi of n (pronounced "zye of en"; symbolized as ξ
). Above 7, the formula produces more reliable estimates. (Note: for those unfamiliar, the sample standard deviation, SD or X-bar (an X with a horizontal line atop it) is an attempt to estimate what the population standard deviation will turn out to be in the future when all such rounds that ever will be fired in history are measured. Population standard deviation is symbolized by the Greek lower case letter sigma (σ).) To better determine SD values for a sample that is 7 or fewer, multiply the ES by ξ
, where n is the sample sizes:
Code:
n ξ(n)
2 1.128
3 1.693
4 2.059
5 2.326
6 2.534
7 2.704