Archery is Not Normal

    The perfect archer would hit the center of the target at every distance with every arrow.   I proudly display a pair of "Robin Hood" arrows that I shot while tuning a new bow.   The second arrow hit the first arrow exactly in the middle of the nock and split the first arrow perfectly down the shaft.   I was thrilled, my new bow was fantastic, I was a great shot, and I have not repeated this level of shooting precision after many, many, many attempts.   

    What is the expected arrow shot pattern for an average, good, and exceptional archer?   What does the grouping of five, ten, and one hundred arrows tell us about our ability to reproducibly shoot a bow?   I will argue that even the best archer shoots fewer arrows in the center of the target than they should if errors in shooting were truly random - IN ONE DIMENSION.   In statistical terms, I will argue that our archery shot patterns are NOT NORMAL relative to the a typical error distribution.  In this post I present some background data on the expected normal distribution of shots, the experimental data that shows that we really shoot a "doughnut" or in some cases an "inner tube" of shots as we attempt to hit the center of the target, and finally, present a reasonable error model that explains the actual shooting results.

    A Normal or Gaussian distribution describes the probability distribution of a large number of independent random events.   A good example might be the average height of adult archers.   If you measured the height of 100 archers and created a histogram of the heights you would get a histogram that looks like Figure 1.  The largest bar in the histogram would be at the average height of the group with decreasing numbers of shorter and taller archers as you move away from the average height.   The spread in the archer heights defines the standard deviation in the data.   A large spread in heights would make the histogram plot in Figure 1 short and broad, while a narrow spread in heights would make the histogram plot tall and thin.   The standard deviation of he data describes the width of the bell curve that fits the shape of the histogram plot.  The Bell Curve is useful because the area under the curve gives the probability that the next archer will be of a certain height. For example, the probability of finding an archer with a height between 69 inches (the average)  and 100 inches is 50% since this range includes half the area under the curve.  The probability of being between one standard deviation below the average height and one standard deviation above the average height is 64.2% or the area between the two green arrows Figure 1.   The normal distribution is a powerful tool for estimating the probability of future events if you have enough data to calculate the bell curve.  Statistics will never give us the absolute answer, but instead allows us to anticipate the result of a new measurement in terms of well defined probabilities.   This is why casinos keep making money.   They are 99.9 % certain that more gamblers will lose money than will win money.   

Figure 1. Normal Distribution or Bell Curve of Archer Heights.

    How does this apply to archery?   No matter how good the shooting form, every archer knows that it is very difficult to keep the sight pin exactly on the middle of the target.   The bow moves as we take aim and shoot.   We also introduce errors through small variations in our anchor points, torquing the bow grip, and shooting during gusts of wind.  If all of these errors were completely random we would expect these errors to produce a distribution of shots with a normal distribution to the error.   The most shots should be in the middle of the target with a distribution of hits away from the center defined by the standard deviation in our shooting.  A simulation of the expected normal distribution in shot placement is shown in Figure 2.

Figure 2. Normal Distribution of shots (red circles) computed with a standard deviation of 3.  This means that 29% of the shots should be inside the 10 ring, 50% of shots should be inside the 9 ring, and 65% of shots should be inside the 8 ring.  The solid line is the bell curve for this shot data based on the absolute distance of each shot from the center.

    This type figure is exactly what is shown in almost all introductory science texts when they are discussing uncertainties in experimental data using an archery analogy.   It is now clear to me that text book authors must not be archers.   My actual shot patterns are shown in Figure 3.  My shots don't cluster in the middle.  I do have some bullseye shots, but many more of my shots cluster around the bullseye than would be expected by a normal distribution of errors.   To my eye my shots seem to cluster in the shape of a donut.

Figure 3.   Shot patterns for five different targets.  Each target was shot with over 25 arrows. 

    Do other archers shoot the same way?   Figure 4 shows the shooting results of Thomas Liston as reported in his book the Physics of Archery.  Liston recorded the distance from the bullseye to each arrow for over 2000 shots!  His data clearly doesn't cluster at the center, but instead has a peak at 2 units from the center.   

Figure 4.  Shooting Results of Liston.   Data is from the Physics of Archery
 augmented with additional shot data from the author.

    But what serious archer would trust the shooting data from a chemist and an engineer?  I decided to test my "donut" shot pattern observation against data from competitive archers.  Archery results posted on the web don't list the distribution of shots, but do list the total score, 10-ring, and X-ring shots.   If you know the total score, and assume the shot pattern follows a normal distribution it is possible to predict the number of 10-ring and X-ring shots.

Figure 5.   Predicted shot distribution for a standard deviation of 3.  The area under any portion of the curve is the probability of shooting the next arrow in that "ring".

    As shown in Figure 5, a simple normal distribution of shots with a standard deviation of 3 predicts that 13.2% of the shots will be in the X-Ring, 26% in the 10-Ring, 18.8% in the 9-ring, etc.  From this data it is possible to predict the total number of arrows in each ring and the total 144-arrow score.  The smaller the standard deviation the tighter the error curve and the higher the predicted score.   Interestingly, this approach always predicts too many arrows in the X-Ring and 10-ring, and too few arrows in the 8-ring and 9-ring (Figure 6).

Figure 6. Predicted and Actual shot numbers in the X-ring or 10 Ring as a function of total 144-arrow score.

    Notice that the actual data points are all below the predicted line based on a simple normal distribution of shots.  For shot totals approaching 1400 the archer must be hitting the 10-ring with almost every shot, but still has fewer than expected shots in the very middle of the target.   My conclusion is that most archers are shooting a "donut" around the middle of the target with less 10-ring and more 9-ring shots.

    I have been thinking about this observation for several months and came up with a range of exotic explanations.  My favorite was that an archer oscillates the sight point back and forth across the target and that the pattern of shots is related to this oscillation with the most probable shot at the edges of the oscillation.   This model would produce a shooting donut.  However, their is a simpler model, and in this case, I will argue that simpler is better.

    Any archery target at a known distance is effectively a two dimensional object.  One dimension is in the horizontal and the other dimension is in the vertical.   It is reasonable to assume that shooting errors in the horizontal and vertical are not correlated (independent of each other).   Therefore,  when considering the error distribution of shots we really should consider the error distribution in two dimensions instead of one.  A technical search of the literature produced an paper by M. Brown, 1963. A Generalized Error Function in N Dimensions, Technical Memorandum #NMC-TM-63-8 from the U.S. Navy Missile Center.  These folks are definitely in the business of hitting the target!  The mathematics is a bit complicated, but the idea is fairly simple - errors in the X and Y directions are independent of each other and when these errors are added together the error in shot placement is very different than a simple normal distribution.   Remember that your score in archery is based on distance from the target center.   This is the hypotenuse of the X distance and Y distance, and both the X and Y errors are independent.  If you shoot with a standard deviation of 1 in the X direction 64% of the arrows will hit within one unit distance along the X axis.  However, a real shot has a Y component that also has error.  This increases the arrow distance from the center due to the Y component.

                arrow distance = (X2 + Y2)0.5

Figure 7 shows the predicted shooting results for 2000 arrows for a standard deviation in X and Y of 1.  Clearly you can't shoot a negative distance, but I plotted the results on both sides of zero to illustrate the shape of the shooting donut.  Unlike a one dimensional error distribution with a peak at the mean, this two dimensional error curve has a peak at one standard deviation from the mean.

Figure 7.  Predicted shooting results for 2000 arrows showing the distribution of shots from the center with a standard deviation in both the X and Y direction of 1.

Figure 8 shows the predicted shooting results for 2000 arrows for a standard deviation in X and Y of 3.  The bigger the standard deviation the bigger the donut and the smaller the standard deviation the smaller the donut.  

Figure 8.  Predicted shooting results for 2000 arrows showing the distribution of shots from the center with a standard deviation in both the X and Y direction of 3.

    Notice the general similarity to Figure 4, the shooting results from Liston, which also shows a peak in shot density at a distance of 2 from the target center.   Figure 9 shows the same data as it would appear on a typical archery target.  The highest density of shots is located the 8 and 9 rings of the target consistent with the peak in figure 8.  

Figure 9.  Shot pattern for 2000 shots with a standard deviation in both the X and Y direction of 3.

The Excel spreadsheet used to perform these calculations is attached to this post so you can perform the analysis yourself.

    It would also be possible to shoot a very tight group with the donut inside the 10-ring, so this analysis doesn't preclude a perfect score, but it does make perfect scores more technically demanding.  Of course, it is not expected that the error (standard deviation) in the X (horizontal) and Y (vertical) direction will always be the same.   A particular archer's shooting form could make them more precise in one direction or the other.   If you know the exact location of each arrow, it would be possible to calculate the distribution of errors in only the X or Y directions to answer this question.  For the technically inclined, several iPhone apps will record you shot positions for this type of analysis.

    Finally, I reanalyzed the shot data from the competitive archers to see if the number of X-ring and 10-ring shots were consistent with a two dimensional error distribution.

Figure 10.  Actual plotted against predicted 10-ring and X-ring shot numbers for competitive archers shooting 144 arrows.  The solid line has a slope of one.

The data still has some scatter, as expected for real data, but the number of 10-ring and X-ring shots are in good agreement with a two dimensional error model.  This isn't proof of any shooting error model, but it is useful that a simple 2D error model can explain all of the major features of arrow shot distributions.

    I sent my first draft of this post to Thomas Liston for his comments on shot distributions. After all, it was his careful record of shot placement for over 2000 arrows that got me thinking about the distribution of shots in the first place.   He replied that a better indication of shooting precision is shot density or the number of shots that hits a specific area of the target.   Shot density can be computed by dividing the number of shots in any distance band from the target center by the area of the band.   The area of the X-ring is the smallest and expands rapidly according to the formula:

Area = Pi(R2out - R2in)

where Rout and Rin is the radius on the outside and inside of any distance band.  Figure 11 shows the shot density for Liston's shooting data.

Figure 11. Liston's shooting data with calculated shot density in green.

The shot data peaks in the center of the target.   Liston points out that if you were a bug on the target the green line in Figure 11 proves that the center of the target is the most dangerous place to land!   I agree, but also contend that the center of the target would be even more dangerous if we didn't shoot consistent with a two dimensional error model (the donut).   I also argue as a turkey hunter that we are not shooting at a small bug, but rather a target the size of kiwi or orange.   Take a look at the predicted 1000 shot pattern for a two dimensional error distribution in Figure 12.   A kiwi anywhere inside the 9 ring would have been hit by an equal number of arrows, due to increased total number of shots between the 10 and 9 rings.

Figure 12.  Simulation of 100o shots with a 2D error distribution.

The middle "area" of the target is still the most dangerous, but not as dangerous as you would expect based on a simple 1D Gaussian distribution of shots, and the bug located anywhere inside of the 9 ring is equally dead.  The last point is the nonintuitive result of the model and the reason I like to build models.   Of course, bow hunting is really a 3D game if you include the error in estimating range.  That is why I added the Gaussian Z column into the model.   For now I have left this error small (I use a laser range finder).  The distance error calculation will be a bit tricky since it will only add to the target error in the Y dimension and, as you may recall from previous posts, the relationship between target error and distance error is not linear.


    Analysis of archery shot patterns is consistent with a two dimensional error model where the archer's shots have normally distributed errors in both the horizontal and the vertical.  These errors are independent of each other and follow a typical Gaussian pattern centered on the target center.   The combination of the errors results in a shot pattern in the shape of a donut where maximum shot density occurs away from the target center leaving a hole in the shot density at the target center.  The size and depth of the hole is related to the standard deviation of shots in the horizontal and vertical dimensions.

Whitney King,
Oct 4, 2012, 8:13 PM
Whitney King,
Aug 23, 2012, 7:25 PM