Grouping is a measure of how well you are able to reliably bunch arrows together. There are many parameters that determine group size, including both the archer and the equipment.
Fig 1. Group size changes over ranges with
complicating factors. The horizontal
scale is significantly exaggerated.
As the equipment is progressively tuned, the grouping approaches the ideal where the prime determinant of group size becomes the archers form.
The distribution of arrows on a target is generally accepted as being a
two dimensional normal (or Gaussian) probability density distribution -
the classic bell curve. The following diagram illustrates the normal
probability density distribution in one dimension: 
Fig 2. Probability distribution of arrow's impact radial distance from target center for archers of different skill levels.
The width of the bell curve is measured in terms of the standard deviation denoted by the Greek sigma character σ. A key feature of the normal distribution curve is that it has a well-defined and predictable shape. This means that the probability of an arrow striking within a certain distance from the target center can be determined with precision if the standard deviation is known.
|
Spread in standard deviations
±1σ |
Proportion within zone 68.27% (~4 out of 6) |
Assuming the sights are properly aligned, the target center has the highest probability of being hit, but for poorer archers this probability is relatively low and not much different to nearby areas.
In FlyingSticks' two dimensional target, with the lower scores occupying a progressively larger annulus areas, the probability is integrated over the target face, assigning a score probability and finally a score estimate. FlyingSticks does this analysis numerically - it will calculate your most likely round score from your group size and in reverse it can calculate your group size from your round score.
In all likely hood, your vertical and horizontal grouping will be different - although for better archers the grouping will be more or less the same in both directions. This occurs as the error sources can be different and largely independent. For example, torqueing the bow grip will result in an increased horizontal spread with little change in the vertical.
Often it is assumed the grouping sized will increase directly proportional to range, but this is not necessarily true (refer to Fig 1.). Wind, canter and some arrow aerodynamic effects have an impact that is more proportional to the square (or other power) of range. The result is if you double the range, the group size will more than double. FlyingSticks euphemistically refers to this as the "Square Law", knowing full well that it is more involved, but none the less useful.
For convenience, FlyingSticks defines an archer's grouping performance in
terms of an oval (superimposed on a target) in which 5 out of 6 arrows can
be expected to land and one land outside the oval. This definition assumes
an arrow of zero diameter.
Fig 3. Group Oval to achieve 5 in 6 shots landing in the group. The faint rectangle shows a simple rectangular group distribution with the same 5 in 6 shots falling within the group.
where σw and σh are the archer's width and height standard deviations at a particular range. Note that the width and height of the oval is 3.58 standard deviations. This figure provides a correction from a one dimensional probability density distribution (represented by the sides of the rectangle - 2.77σ) to the oval form.
This is an arbitrary statistical definition chosen simply because 6 is a
frequently used number of arrows per end and it has a degree of
intuitiveness. Illustrated as a one dimension bell curve, the group width
and height looks like:
Fig 4. Arrow group width shown graphically, with 5 out of 6 in the gold. For ethical hunting the group size is increased to approximately ±2σ, requiring 21 out of 22 to be within the group.
For the technically minded, FlyingSticks calculates the standard deviation for the internal shot simulator from a simple relationship to the (5 in 6) group size:
σ = 0.2793 D
where D is the group width or height.
The military have great interest in the subject, so the following are used in that arena. All tend to assume a circular bivariate normal distribution.
This is a term that defines a circle radius centered on the target in which 50% of strikes can be expected. A circular bivariate normal distribution is assumed so allowing other probability radius to be calculated.
A simple root mean square of error all strikes. If a circular bivariate normal distribution is assumed, then approximately 65% of strikes occur within this radius.
Assumes a circular bivariate normal distribution, and is the circle radius in which 95% of strikes occur.
Note that our group width does not define the limits of the group but rather the grouping tendency. Six arrow should be regarded as a minimum. There is absolutely not point in being too dogmatic about these figures for small numbers of arrows - better to be sensibly subjective. With just six arrows it is entirely possible that the group center and the group size be in error by 30% or more.
As the number of arrow increases, the guidelines can be applied with more rigor. Ideally at lease a 36 arrow round is required, but then simply entering the round score is probably simpler. (You could use both methods to confirm your judgement).
FlyingSticks allows you to enter an horizontal and vertical group dimensions and it will calculate the skew factor.
A good way to inform FlyingSticks of your group size is to enter a FITA 1440 round score. This involves 144 arrows so it is a reasonable statistical sample over four different ranges. FlyingSticks does a numeric reverse probability analysis. When the archer recognizes that his grouping in the vertical is different to the horizontal, there is an opportunity to enter a skew factor.
If you wish, you can break down the round into the individual range scores with a more comprehensive analysis.
If you do not have a round score, you can still visually estimate the group size. It turns out the standard deviations of a group can be intuitively estimated with reasonable accuracy once you understand how FlyingSticks defines a group.
Interestingly the normal distribution makes no assumption all arrows will be within a certain area. Any bad score for a shot is possible but become increasingly unlikely for better archers. However, if an archer's form were perfectly normal, then theoretically only a one in a million shots will be outside a 10σ wide oval. In practice a shot can be way of the mark due to some mistake. Such shots should be excluded from the statistics.
There is some evidence that archers lacking well developed form may not have a normal distribution or may have multiple means, each with its own distribution. This can repeatedly result in more than one distinct group. This may be caused by distinctly difference release techniques, bow torqueing, cantering etc. As form improves the less dominate "modes" are gradually eliminated.
Statistical variation plays an important role in WA rounds due to the relatively small number of shots involved. You can be lucky or unlucky!
Fig 5. FITA 10 Zone Face
FlyingSticks maintains you grouping details as a record of your personal progress over time. The information is used to recommend when form adjustments or bow tuning may help. The impact of some adjustments may be small relative to your group size, so trying to tune them is near impossible because the improvements are swamped by poor form.
For hunters the performance information is one of several factors in determining the boundaries for ethical shooting.