Archery Geekery

Since the new UK handicap and classification systems were released last year, I’ve had a steady stream of enquiries from fellow archery geeks wanting to understand a bit more about the maths of how the system works. So, this will be the first article in a series that will explain all of this.

The UK archery handicap system is a mathematical and engineering marvel which is the envy of archers around the world (this is not an exaggeration – archers from many countries enquire routinely about how they can adopt it). This system was designed and developed in the 1970s by the late David Lane, and was maintained by him for over 40 years. Whilst the outward facing structure of the system remained largely the same over that time, the inner workings and equations inside were subject to regular adjustments over the years to better model the changing nature of the sport, with each new update changing the tables of scores required at differrent performance levels. The last major update was in 1995. In an open letter to the archery community in 2015, David offered to hand over the stewardship of the system to the next generation of archery geeks. Some conversation with him followed and over the course of 2 years from 2021-2023, a team of expert volunteers, using input from Lane and others, tweaked the system to fix some longstanding issues and ensure its continued longevity for the foreseeable future.

This article will first describe Lane’s original scheme, and then highlight some of the slight adjustments that were made in 2023.

What is a handicap?

The handicap system is a set of mathematical equations that describe two things –

How the shape and size (the distribution) of the an archer’s arrow group varies with their skill level.
How that distribution of arrows would translate to specific scores on a given target face at a given distance.

In mathematical terms we would describe this as a model – a mathematical description of the factors and their relationships that affect the groupings of arrows, such that we are able to make accurate predictions about them. Like many models, this one has both theoretical parts based purely on understanding of the physics of how projectiles fly on planet Earth, and empirical parts where we model the actual behaviour of arrows in the sport of archery.

Arrow Group Distribution

First, imagine a perfect archer, with a perfectly consistent shot – what that really means is that each arrow would leave the bow at the exact same angle relative to the target each time. Assuming no other external factors, the arrow would land in the exact same spot each time (with each arrow destroying the previous one…). Less skilled archers would not have such good angular consistency, with arrows leaving the bow at different angles and landing in different places. At the simplest level, this measure of angular consistency is what a handicap is.

There are of course other factors, aside from the archer’s angular consistency that affect where the arrow lands (aerodynamics, wind, technique, equipment etc.) which we will come back to later.

The first part of Lane’s equations is a scale of angular consistency from very good (but not perfect) to not so good, and this gives us our basic handicap scale. This is modelled on a logarithmic scale, which is then scaled down.

σ_{θ} = {1.036}^{(H + 12.9)} \times 5 \times 10^{- 4}

In this equation, H represents the handicap value. The shape of the the curve is defined by the first term and represents a 3.6% improvement in group size per handicap point. This scale is relative to the baseline anglular deviation (5×10^-4). The end result σ_θ is represents the angular deviation measured in radians. For those that are more familiar with degrees rather than radians, that means a handicap of 0 represents a level of skill where arrows are all launched with an average deviation of 0.045º, whereas a handicap of 100 means arrows all leave the bow with an average deviation of 1.56º. We can plot that curve to see how the angle changes through the handicap range –

Making a big assumption that arrows fly in a perfectly straight line we can calculate what actual distance from the centre of the target that angular deviation would create at a particular distance. For instance, this chart shows what that would look like at 70m. We can see that above a handicap of 87, the arrows would be more than 122cm displaced on average i.e. missing the target.

This displacement is represented in Lane’s equations by scaling up the angle – multiplying the angle in radians, σ_θ, by the range to the target (R) and converting into centimetres (x100) to give this measure of σ_r, the displacement in cm on a target at range R.

σ_{r} = 100 \times R \times {1.036}^{(H + 12.9)} \times 5 \times 10^{- 4}

If we lived in a gravityless vacuum, that equation alone might be enough. In the real world though we know that arrows don’t travel in perfectly straight lines and that there are things that affect the flight of an arrow proportionally to how long it’s in the air (i.e. by the range to the target). This measure of how the group size is affected by range is know as the excess dispersion. Deciding how to model this excess dispersion is where the handicap system starts to become a bit more empirical than theoretical. One of David Lane’s papers describes his approach –

The problem which has to be solved for the construction of handicap tables is to find out how much the average angular error increases with increased target range and whether the same relationship applies to both sexes, all age groups and all standards of archers.

Lane considered examples from high-level tournament archers, and from less experienced archers, to find an approximation that worked for the whole spectrum. Things were quite different back in the 1970’s though, both in the standards of shooting, and in our ability to do large scale data analysis. This part of any handicap system requires us to fit a model to available data in order to minimise the errors over the range. There is no perfect answer. As arrow speed and drag is a determining factor of how much excess dispersion there is, the system also needs somehow cover a range from high-speed compound bows down to lower-speed longbows or low-poundage juniors. There are a lot of other factors and assumptions at play here that we will examine in a future article.

In the end, Lane took a shortcut of recognising an generally true correlation between arrow-speed and handicaps – that people with higher arrow speed generally shoot higher scores and vice-versa. However, this is a quite a big assumption and untrue in many cases. This is part of the handicap model is a bit of a mathematical workaround and one that lane himself required “a great deal of trial and error to make sure that the resulting tables provided a reasonable match to actual shooting results“. His final expression for the excess dispersion was –

F = 1 + 1.429 \times 10^{- 6} \times {1.07}^{(H + 4.3)} \times R^{2}

Where 1.429×10^-6×1.07^H+4.3 is factor that relates handicap ratings to arrow speed, and R² represents the fact that excess dispersion increase with the square of distance.

The final equation for deviation from the pinhole then becomes

σ_{r} = 100 \times R \times {1.036}^{(H + 12.9)} \times 5 \times 10^{- 4} \times (1 + 1.429 \times 10^{- 6} \times {1.07}^{(H + 4.3)} \times R^{2})

This expression can obviously be simplified a bit, but I won’t do that for now as it makes it easier to compare against other systems later. The constant 12.9 was chosen to shift the entire scale so that a handicap of 0 produces a score of 1400 on a 90m WA1440 round, this is the so called scratch point, and is not mathematically significant, but is more a matter of policy about where we want the range to max out.

If we plot this new version of σ_r, we can see a slightly more pessimistic displacement now. The added excess dispersion term shows displacement increasing more rapidly through the handicap range, and shows a more realistic position where archers around 65 handicap will be regularly missing the target at 70m.

However, this graph also hints at one of the problems with the original handicap models. The displacement rockets up extremely quicly, becoming near vertical quite early in the handicap range and this is one of the reasons why this system does not fit well to low-poudage or novice archers shooting longer distances.

At this point we have an expression that represents the RMS deviation from the pinhole. It doesn’t tell us where the arrow will land exactly, but it describes the distribution of where we expect arrows to land. The next job is to figure out what that distribution would look like on a particular target face.

Scoring Functions

The σ_r expression which represents the RMS deviation from the pinhole can be converted into a probability distribution of the form

p (r) = \frac{2 r}{σ_{r}^{2}} \exp (\frac{- r^{2}}{σ_{r}^{2}})

We then need to “overlay” that probability distribution on a target face, to estimate which scoring rings arrows in that distribution would land in, and therefore what their average score would be. These can be represented by step or staircase functions S(r), which we integrate over –

\bar{S} = \int_{0}^{\infty} S (r) \cdot \frac{2 r}{σ_{2}^{r}} \cdot \exp (\frac{- r^{2}}{σ_{2}^{r}}) d r

The S(r) functions are different for each type of target face. For example, a simple 10-zone metric scoring function looks like this

\bar{S} = 10 - \sum_{1}^{10} \exp (\frac{- {(\frac{n D}{20} + 0.357)}^{2}}{σ_{r}^{2}})

Key things to note with this function –

This loops over each scoring ring (n) of a target face with diameter D.
Calculates each ring radius as nD/20 (so for example the radius of the 2nd ring (the 9-ring) on a 122cm face is (2*122/20)=12.2cm
The size of that radius is slightly expanded by half the diameter of a typical arrow (0.357cm) to account for line-cutters.
At each scoring ring it’s subtracting the expected score loss from that ring from the maximum.

Scoring functions are available in David Lane’s paper (link below) for most standard target faces, and others can be worked out quite easily. For example, here’s a function for a 5-zone triple spot, with compound-style inner-10 scoring. In this function we see the first scoring zone at 1/40th of the face diameter (not the normal 1/20th), the next 4 zones at normal 1/20th diameter, and then all score from what would be scoring zones 5-1 is lost.

\bar{S} = 10 - e x p (\frac{- {(\frac{n D}{40} + 0.357)}^{2}}{σ_{r}^{2}}) - \sum_{n = 2}^{4} \exp (\frac{- {(\frac{n D}{20} + 0.357)}^{2}}{σ_{r}^{2}}) - 6 \cdot \exp (\frac{- {(\frac{5 D}{20} + 0.357)}^{2}}{σ_{r}^{2}})

The very last step is to multiply all of this up to give a score for a whole round, made up of m parts of N_m arrows at range R_m on a face of diameter D_m, using the appropriate staircase function S for that face. For example, for a 1440 round we would do this 4 times with 4 different distances and two different face diameters

S = \sum_{1}^{m} N_{m} \cdot \overset{―}{S} (R_{m}, D_{m})

Let’s try this out and create a handicap table. To save space, we’ll not calculate the full table from 1-100, we’ll do it in steps of 5.

This table shows all the essential elements of the handicap model coming together – first the calculation of σ_r down the handicap scale, then the calculation of the score-loss components of the staircase function (10 steps in the case of this 10-zone target) to give the average score per arrow followed by a final multiplication by the number of arrows in the round. The end result in the final column is the handicap table (an abridged version in this case, but the same method is used to calculate any sized steps)

For other rounds which have multiple different distances in them, we just repeat this calculation for each distance as if it it were an independent round, and then add them all together for the final score. The code for handicap table calculation on this website, does exactly this.

This is everything you need to know to create a David Lane era handicap table. Now let’s look at the changes that were introduced in 2023. You’ll be surprised how small these changes are. In fact, to change the code that runs this website to use the new system took about 60 seconds…

The 2023 Handicap System

As I mentioned earlier, the Lane model was updated in 2023. This addressed some well-known issues in the previous system where the handicap system was under-predicting short-range scores and over-predicting long range scores for people at the lower end of the handicap range, and was doing the opposite for archers at the higher end of the spectrum. Also since, the previous update of the system, equipment had evolved, standards increased, and crucially, large volumes of tournament data were now easily accessible to allow the excess dispersion to be modelled more completely.

After nearly two years of work, updates to the handicap system were released –

The excess dispersion is now modelled different with an exponential term after this was found to be a better fit than the previous model to real data, right the way from World Cup events to local county tournaments.
The range of the handicap was expanded from 0-100 to 0-150, to allow finer granularity in some areas of the scale, without sacrificing the lower end of the scale often used by juniors and beginners.
The typical arrow diameter was updated to reflect the reality of modern archery, and different diameters are now used for outdoor and indoor staircase functions. An arrow radius of 0.275cm is used for outdoor rounds and 0.465cm for indoor rounds.

What does this look like in practice? In reality the change is just in the definition of σ_r which now looks like this

σ_{r} = 100 \times R \times {1.035}^{(H + 6)} \times 5 \times 10^{- 4} \times \exp (0.00365 \times R)

If we plug in that new equation and repeat the table generation exercise above, we get the following table –

And that’s it – that’s how to generate handicap tables for any round in the new handicap system. At this point, you’re probably thinking – “that’s a bit underwhelming!” or “that’s not much of a change” and you’re right! The new system is not dramatically different from the old one, and that’s deliberate. The old handicap system was based on extremely solid foundations, and worked very well in a lot of situations. The new equations are of the same general form as the original ones but now have an excess dispersion term which more closely relates to real-world data.

In fact, when you look at a handicap system, whether it’s the ArcheryGB system (either the new one or the old one), or the Archery Australia system, we find that they are all quite similar. They all have

A term that describes the distance
A term that describes the group size at handicap 0
A term that describes how that group size varies with handicap
A term that describes how that group size varies with distance

		Distance term	Variation of group size with handicap	Angular deviation at H=0	Variation of group size with distance
Old AGB System	σ_r=	100 x R	${1.036}^{(H + 12.9)}$	5 x 10^-4	$1 + 1.429 \times 10^{- 6} \times {1.07}^{H + 4.3} \times R^{2}$
New AGB System	σ_r=	100 x R	${1.035}^{(H + 6)}$	5 x 10^-4	$\exp (0.00365 \times R)$
Archery Australia system	σ_r=	100 x R	${0.997}^{(- H - 0.0064)}$	1.41 x 10^-3	$\exp (0.004 \times R)$

When seen like this, we can see a great deal of similarity between the systems, and it is reassuring to see independent analyses coming to the same general conclusion.

Comments

One response to “Handicap System Maths”

Rob

January 5, 2025

Thanks, this is very interesting and prompts a few questions that you may already be considering for a future article:

– could you provide more information on the data analysis, maybe some distribution histograms of calculated handicaps for the data used would be interesting?
– can you say more about the correlation between draw weight and handicap? e.g. plot handicap vs. draw weight datapoints for recurve at 70m

The link to Jack Atkinson’s python code answers a lot of technical questions.

Handicap System Maths

What is a handicap?

Arrow Group Distribution

Scoring Functions

The 2023 Handicap System

Further Reading

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One response to “Handicap System Maths”

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