Calling all math experts. I need some help to find a way to mathematically shift hole scoring distributions up or down to predict what they will look like if players of higher or lower skill play the hole. Let's say that a group of players averaging 950 play a hole and the scoring distribution looks like this:

20% 2s
55% 3s
25% 4s
Avg = 3.05

I have the formula to predict the change in scoring average among different levels. We know that a group of 1000 rated Gold players should average 2.77 and a group of 900 rated White level players should average 3.33 on this hole. However, what will the percentage of 2s, 3s and 4s look like for Gold & White? I can probably eyeball the numbers taken some time but that's tedious when doing much analysis. Is there a mathematical process here that I'm overlooking?

This is important for our designers who are testing holes designed for a specific player level but may not have a large enough group at that level playing in an event. So they have to adjust the distributions they get from (usually) lower level players to estimate how it will play for the intended level.

The example distribution shown is fairly simple but it can get more complicated. In some distributions, Gold level might have 15% 2s and no 5s on a hole. But by shifting the curve for White level, they should have no 2s and some 5s. It's not clear at all how I can do this mathematically as opposed to some human iterative adjusting process.

Golly. Chuck asking for math help. Isn't that like Hercules asking a redcap for help with his luggage?

Anyway, a simple method is just to assume a Poisson distribution with a mean - and thus standard deviation (Or is it variance? Doesn't matter.) - equal to the expected average score. Adjust the average score (mean) up, and the spread also gets wider, the percentage of low scores gets smaller and the percentage of high scores gets bigger.

For a rough "smell" test, you could see how well the Poisson predicts the observed frequency of each score, given the average, for a bunch of hole and skill combinations.

That should get you by until I have time to think about it.

I've figured it out using the NORMDIST function within Excel that returns the area under a normal curve if you know the Average and Std Dev. Although the actual scoring distributions for holes are not always ideal "normal curves," it appears to work pretty well. It's interesting seeing what the odds of an ace are on certain holes.

I don't know. What if an ace is impossible, like on a dogleg? Does the math show that it's possible?

Probablility should be zero when the possibility is zero.

For purposes of forecasting, I've rolled the ace probability into that for a 2 to avoid oddities such as aces on par 4 doglegs, even if it was equivalent to winning the lottery. Just for fun though, I looked at what they might be on some legitimately aceable holes.

1. Using Normal, if the mean is, say 2.5, your model would tell you that the probability of a 6 is the same as the probability of a negative one. That ain't right.

2. Since the first throw is a given, it adds no information, so what you really want to be modeling is the number of Extra throws.

3. Poisson has no left tail that goes left beyond 0 (corresponding to an Ace), and the right tail is skewed farther to the right (corresponding to, say, my score).

4. Using normal, you have to make an assumption about how the standard deviation changes as the average score changes. Intuitively, it seems it should widen as players get worse. With normal, you have to guess how much. With Poisson it changes automatically.

Switching to Poisson and modeling Extra throws should be a minor modification from where you've already done, and you won't have to deal with nonsense results at the extremes.

Of course, none of this takes into account the actual observed distributions, which is a shortcoming. There may be something about a hole that makes a two very unlikely, but an ace or a three common. Perhaps a short hole on an island.

A way to take this into account is to use stacked binomial distributions. What is the observed probability of the first throw going in? Given that the first throw didn't go in, what is the probability that the second throw goes in, etc?

Then, to increase the mean, you just reduce the probability of each throw going in. To decrease the mean, you reduce the probability of each throw NOT going in. The simplest way would be to multiply all probabilities by a single factor. Solve for the factor that produces the target mean.

This method adjusts your example as follows:

Mean = 3.05

1 0
2 20
3 55
4 25
5 0
6 0

Mean = 2.77
1 14
2 15
3 52
4 19
5 0
6 0

Mean = 3.33

1 0
2 16
3 46
4 31
5 6
6 1

For that hypothetical island hole, it would do this to a concocted distribution with mean 3.05.

Mean = 3.05

1 10
2 5
3 55
4 30
5 0
6 0

Mean = 2.77

1 18
2 11
3 48
4 23
5 0
6 0

Mean = 3.33

1 8
2 4
3 46
4 34
5 7
6 1

A further refinement would be to use different adjustment factors for each throw. For example, on a 950 foot uphill hole, you might want to leave the probability of the first throw NOT going in at zero � no matter how good the player.

I think a case might be made that the scoring spread (SD) on a hole might be wider the lower the skill level. I'll be monitoring scores at Highbridge from relatively uniform sets of skill levels and see if that bears out. If so, the Std Dev can also be tweaked up or down and still use the NORMDIST function.

The primary purpose of this exercise is not the precision of percentages for each score, but to uncover holes where there might be a narrow range of scores for some level(s). A hole where a specific score is thrown by 80% of the players is not very good for spreading scores. If one is discovered from actual scores, then maybe it should be changed or perhaps it would be OK for another skill level. Likewise, a hole that's OK for the skill level who threw it, might have a too narrow scoring range when these formulas adjust the percentages for another skill level.

Another approach, as any good scientist would use, is to take the observed data and make the theory/formulas/method fit that.

In the case of predicting score distribution of one skill level based on that of another skill level, I'd say it's pretty much impossible. Impossible like stuffing a Cadillac up your nose impossible.

Short furry wise man once say: Course design is an art, not a science.

Data follows.

Walnut Ridge, pre-shuffle pool A (approx 1000 rating) and post-shuffle pool G (approx 942 rating). Here are 5 holes that have similar score distributions in the A pool, and the same holes in the G pool. Holes 1 and 3 are pretty darn close. Holes 4 and 6 are similar to each other, and not all that much different from 1 and 3. Hole 11 is glaringly different.

<table border="1"><tr><td>Walnut - A Pool</td><td></td><td></td><td>
</td></tr><tr><td>Hole</td><td>1</td><td>3</td><td>4</td><td>6</td><td>11
</td></tr><tr><td>Birdies</td><td>34%</td><td>33%</td><td>34%</td><td>33%</td><td>33%
</td></tr><tr><td>Par</td><td>60%</td><td>60%</td><td>58%</td><td>63%</td><td>60%
</td></tr><tr><td>Bogey</td><td>6%</td><td>7%</td><td>7%</td><td>4%</td><td>7%
</td></tr><tr><td>Double</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Worse</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Ave</td><td>2.72</td><td>2.75</td><td>2.73</td><td>2.72</td><td>2.75
</td></tr><tr><td>Stdev</td><td>0.57</td><td>0.59</td><td>0.59</td><td>0.55</td><td>0.59
</td></tr><tr><td></tr></td></table>

<table border="1"><tr><td> Walnut - G Pool</td><td></td><td></td><td>
</td></tr><tr><td>Hole</td><td>1</td><td>3</td><td>4</td><td>6</td><td>11
</td></tr><tr><td>Birdies</td><td>18%</td><td>18%</td><td>22%</td><td>25%</td><td>24%
</td></tr><tr><td>Par</td><td>69%</td><td>71%</td><td>67%</td><td>63%</td><td>50%
</td></tr><tr><td>Bogey</td><td>11%</td><td>8%</td><td>11%</td><td>13%</td><td>24%
</td></tr><tr><td>Double</td><td>1%</td><td>3%</td><td>0%</td><td>0%</td><td>3%
</td></tr><tr><td>Worse</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Ave</td><td>2.96</td><td>2.96</td><td>2.89</td><td>2.88</td><td>3.06
</td></tr><tr><td>Stdev</td><td>0.59</td><td>0.62</td><td>0.57</td><td>0.60</td><td>0.77
</td></tr><tr><td> </tr></td></table>


Grandview, post-shuffle A pool (approx 1000 rating) versus post-shuffle G pool (approx 942 rating). Same deal. Similar holes in A pool versus how they look in the G pool. Three different comparisons here.

<table border="1"><tr><td> Grandview - A Pool</td><td></td><td></td><td></td><td></td><td></td><td>
</td></tr><tr><td>Hole</td><td>6</td><td>7</td><td>12</td><td>3</td><td>17</td><td>5</td><td>9</td><td>13
</td></tr><tr><td>Par</td><td>4</td><td>3</td><td>4</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3
</td></tr><tr><td>Eagles</td><td>1%</td><td>1%</td><td>1%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Birdies</td><td>65%</td><td>67%</td><td>67%</td><td>35%</td><td>35%</td><td>28%</td><td>28%</td><td>26%
</td></tr><tr><td>Par</td><td>31%</td><td>32%</td><td>31%</td><td>58%</td><td>54%</td><td>71%</td><td>69%</td><td>72%
</td></tr><tr><td>Bogey</td><td>1%</td><td>0%</td><td>1%</td><td>7%</td><td>11%</td><td>1%</td><td>3%</td><td>1%
</td></tr><tr><td>Double</td><td>1%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Worse</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Ave</td><td>3.36</td><td>2.31</td><td>3.32</td><td>2.72</td><td>2.76</td><td>2.74</td><td>2.75</td><td>2.75
</td></tr><tr><td>Stdev</td><td>0.61</td><td>0.49</td><td>0.53</td><td>0.59</td><td>0.64</td><td>0.47</td><td>0.50</td><td>0.47
</td></tr><tr><td> </tr></td></table>

<table border="1"><tr><td> Grandview - G Pool</td><td></td><td></td><td></td><td></td><td></td><td>
</td></tr><tr><td>Hole</td><td>6</td><td>7</td><td>12</td><td>3</td><td>17</td><td>5</td><td>9</td><td>13
</td></tr><tr><td>Par</td><td>4</td><td>3</td><td>4</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3
</td></tr><tr><td>Eagles</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Birdies</td><td>40%</td><td>50%</td><td>33%</td><td>29%</td><td>13%</td><td>14%</td><td>21%</td><td>8%
</td></tr><tr><td>Par</td><td>51%</td><td>46%</td><td>56%</td><td>58%</td><td>71%</td><td>76%</td><td>76%</td><td>86%
</td></tr><tr><td>Bogey</td><td>8%</td><td>4%</td><td>11%</td><td>13%</td><td>14%</td><td>10%</td><td>3%</td><td>6%
</td></tr><tr><td>Double</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>1%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Worse</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>1%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Ave</td><td>3.68</td><td>2.54</td><td>3.78</td><td>2.83</td><td>3.08</td><td>2.96</td><td>2.82</td><td>2.97
</td></tr><tr><td>Stdev</td><td>0.62</td><td>0.58</td><td>0.63</td><td>0.63</td><td>0.67</td><td>0.49</td><td>0.45</td><td>0.37
</td></tr><tr><td> </tr></td></table>


Now the other way around, similar score dist in G pool versus how they look in A pool. Two different comparisons here.

<table border="1"><tr><td> Grandview - G Pool</td><td></td><td></td><td>
</td></tr><tr><td>Hole</td><td>17</td><td>18</td><td>14</td><td>15
</td></tr><tr><td>Par</td><td>3</td><td>3</td><td>3</td><td>3
</td></tr><tr><td>Eagles</td><td>0%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Birdies</td><td>13%</td><td>15%</td><td>40%</td><td>42%
</td></tr><tr><td>Par</td><td>71%</td><td>67%</td><td>58%</td><td>57%
</td></tr><tr><td>Bogey</td><td>14%</td><td>18%</td><td>1%</td><td>1%
</td></tr><tr><td>Double</td><td>1%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Worse</td><td>1%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Ave</td><td>3.08</td><td>3.03</td><td>2.61</td><td>2.60
</td></tr><tr><td>Stdev</td><td>0.67</td><td>0.58</td><td>0.52</td><td>0.52
</td></tr><tr><td> </tr></td></table>

<table border="1"><tr><td> Grandview - A Pool</td><td></td><td></td><td>
</td></tr><tr><td>Hole</td><td>17</td><td>18</td><td>14</td><td>15
</td></tr><tr><td>Par</td><td>3</td><td>3</td><td>3</td><td>3
</td></tr><tr><td>Eagles</td><td>0%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Birdies</td><td>35%</td><td>14%</td><td>71%</td><td>63%
</td></tr><tr><td>Par</td><td>54%</td><td>78%</td><td>29%</td><td>36%
</td></tr><tr><td>Bogey</td><td>11%</td><td>8%</td><td>0%</td><td>1%
</td></tr><tr><td>Double</td><td>0%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Worse</td><td>0%</td><td>0%</td><td>0%</td><td>0%
</td></tr><tr><td>Ave</td><td>2.76</td><td>2.94</td><td>2.29</td><td>2.39
</td></tr><tr><td>Stdev</td><td>0.64</td><td>0.47</td><td>0.46</td><td>0.52
</td></tr><tr><td> </tr></td></table>

Wow, somehow the formatting got hosed between the Preview and the Post. I ain't none too bright with these computer thingies.

For the first comparison, all 5 holes are similar in the A pool.

For the 2nd comparison, holes 6/7/12 are one group, 3/17 are the next group, and 5/9/13 are the final group.

For the 3rd comparison, holes 17/18 are one group, and 14/15 are the other group.

This analysis has nothing to do with how a hole is designed other than identify those that might need help to better spread scores. The data is more encouraging than discouraging. The data confirms that a higher rated pool shoots better scores and, in most cases, has a std dev that's equal to or smaller than the lower pool.


My forecaster generated similar changes in distribution once a factor was introduced that increases/decreases the actual SD from the data to adjust for rating difference. One thing that affects the numbers is how close the hole gets to a par 2. The A pool can only shoot more 2s since the step change from a 2 to a 1 is a higher barrier than going from a 3 to 2. So holes where the scoring average is under 3 for lower average rating will generate a distribution that is narrower (lower SD) for higher rated players more than for higher scoring holes.

Glad I could provide data to help in your endeavor.

Good Luck! (http://www.cadillac.com)

You've always been helpful with this type of analysis. It's both frustrating how sometimes the numbers don't seem to work out the way you expect or forecast. But on the other hand, it's also amazing the many things that can be estimated pretty well. If it wasn't tricky, it wouldn't be as much fun to figure it out.

We always need to carefully consider what we are using the math for.

As your data shows, we cannot use simple models to predict the spread (or even the average) of scores of one class of players based on the results of another class, on a hole-by-hole, tournament by tournament basis.

My model, for example, correctly predicted about 75% of the changes in number of people getting each particular score between the two groups. And that was using the actual change in averages, not the predicted that Chuck's formula would produce. Any reasonable formula that adjusts the average to a new, known, target has to get a lot of the changes correct.

To use a formula such as Chuck's or mine to predict whether a hole will have a weird distribution for a new set of players is not appropriate. Any general formula will not be able to take into account the particular characteristics of the hole.

Imagine a 180 foot long wide open hole where the first throw is over a 150 foot wide canal. Scores of 1000 rated players will be mostly 2's, I would guess. There is no way that you could use this data to predict the absence of 3's that will occur for lower rated players (when the first shot goes in the water).

That's not to say that average- and spread- predicting formuals are worthless. But, they should be used the other way around.

In the above example, the formulas would tell you that as player ratings go down, the number of 2's should be replaced by some 3's 4's and 5's. By comparing the actual results to the predicted you can see the dearth of 3's, and discover that there is something distinctive about that hole.

NIce a thread with math geeks just what I was looking for :D

Chuck please dont take this so personal and get so overdefensive!
You told me 2 years ago that most of the kinks would be worked out by now.
I am sorry to say but they are not and the rating system needs to looked at from a wider angle lense.

Your system has been good to get a reasonably fair starting point for most players, but it needs work.
It also needs to be able to be used to RANK players against each other each week who do not play against each other.

A ranking system should be the overall goal at least at the top level of play!

Here we go:

I have a major problem with the player rating system and it being skewed so that players who play harder courses have lower player ratings. ( I know your adjusting the SSA to correct this) Why didnt you think of this from the start?

It seems as though each stroke is worth around 10 points when the scores are between 48 and 54, but they are only worth 5 when the scores are above 65.
I am no math expert, but wouldnt it make more since to have a constant value of each stroke and a baseline using average scores and player ratings of more players?

Heres the example of what I think is a much easier formula.

If you take all of the scores from the advanced and pro divisions droppping the bottom 10%, assuming they play the same configuration, ( which I strongly reccomend) and averaging theirs scores.
lets say the average was 58!

Then take the same players "CURRENT" player rating and also make an average.
Example average of 980.

This would mean that a 58 would be 980 rated round.
If each stroke was worth 10 than a 50 would be 1060 and a 64 would be 920.

Maybe each stroke could be worth 5 or 7, but shouldnt it be the same for all strokes whether the average is 65 or 54?

If anything points should be worth less for the lesser average scores.

Is there a reason why this will not work better?

I feel the the whole propagator thing is also a problem.
Most likely i dont know exactly how it works though :confused:
If 2 or 3 of the best players or propagators there shoot poorly or GREAT it seems to have too much effect on the player ratings.

Why wouldnt you want to use more data for the base line like the averages of 90 % of advanced and pro field?

I feel it is very important to develop a player RANKING System.
Why do you think they have rankings in Ball Golf, Tennis, college basketball/ football, nascar and other sports.
Its because it is better than ratings and much more interesting to look at for the fans.

Trust me when I tell you that a ranking will increase the competitiveness among the players from coast to coast.

I know the trick lies in the "course ratings", but we need to be heading in that direction ASAP!!

A ranking system should be the overall goal at least at the top level of play!

I have a major problem with the player rating system and it being skewed so that players who play harder courses have lower player ratings. (I know your adjusting the SSA to correct this) Why didnt you think of this from the start?



Ranking systems like Sagarin work well for some sports but not disc golf and even ball golf for that matter, although it's less of a problem. What most don't realize is the underlying math for ratings is very similar to the Sagarin process except we also get an intermediate value as a bonus, called the SSA or course rating. Sagarin assumes the competition field, court, diamond or gridiron is the same for each event being evaluated which is a reasonable assumption for those sports but not disc golf. In DG, we have differences in the number of holes, average lengths of holes and type of terrain. Imagine the squawking if we decided to only do ratings on courses with 18 holes, 5500-6500 feet, no elevation and scattered trees? Not very functional. Our players (and courses) are both rated AND ranked with our system. The top 50 ranked players are on the first page of the ratings stats.

The adjustment we made to the higher SSA course factor was modest and we only slightly tweaked the factor. We couldn't set the factor in the beginning as accurately, not because we thought of it later, but because we didn't have enough data on high SSA courses in 1998 when the system first started. Winthrop Gold and Ozark and Fly 18 had no data and PDGA events reported on spreadsheets. We've come a long way on data management.

No matter what alternative ratings system you concoct, higher SSA courses will have two features: (1) Each throw will be worth fewer rating points, and (2) players will not shoot as high and low rated rounds. You can make up numbers to force the ratings points to be the same at all SSAs but the system will not be valid. I'm not sure how many times this has to be said but I'll continue to say it.

I've used this example before but it doesn't stick for some people. Let's say we have two players rated 1000 and 950. They play 18 holes on a 50.4 SSA course and they shoot their average scores which are 50 and 55. Their round ratings are 1000 and 950 respectively. Each throw is worth 10 points (50/5=10). They play the same course a second time with identical results. However, they submit the results to the ratings calculator as scores of 100 and 110 on a course with 36 holes. Their round ratings still come out as 1000 and 950. However, now the difference in their scores for that 36-hole round is 10 throws but the difference in their round ratings is still 50. The value of each throw is now 5 pts (50/10=5).

Now, imagine that instead of 36 holes, the players played a different monster course that was twice as long per hole as the first course. That means they'll play the same total length as the first course played twice, but now in just 18 holes. You would still expect the better player to win by something more than 5 throws as on the shorter 18, but maybe not as much as the 10 throws he won by playing 36 holes. It turns out that the better player should win by 7 throws if each shoots his rating. In this case, this 70 SSA course ends up having each throw worth 7.14 pts (50/7=7.14).

That's how the real world and the math works. It's not magic. We just had to determine how to do the calculations AND actually validate the calculations with real scores.

No matter what alternative ratings system you concoct, higher SSA courses will have two features: (1) Each throw will be worth fewer rating points, and (2) players will not shoot as high and low rated rounds. You can make up numbers to force the ratings points to be the same at all SSAs but the system will not be valid. I'm not sure how many times this has to be said but I'll continue to say it.



the ratings points do not need to be the same at all ssa's. for the system to be statistically valid the points per stroke should increase as the individual score falls further from the center of the bell curve representing all possible scores at ANY given ssa. of course, this would make skinner's round 1250 or so. :eek:

As enticing as it might be to increase the points reward for less probable scores, it doesn't work in practice. Perhaps it still hasn't sunk in that the reason a score is worth the rating it gets, is that a player with that rating actually averages that score. Skinner got a round rating of 1115 because a (super human) player of this rating would regularly shoot a 41 average on the Darkside. Our round ratings actually reflect reality to the best of our knowledge and results so far.

Without knowing the specific data the NORMDIST function of Excel uses, I would suggest this very close approximation to your initial post.

Use your three points to extrapolate and plot a bell curve. Make your x-axis equal the score while your y-axis is the percentage. At a score of 2 mark 20%, (3, 55%), and (4,25%).

After you draw your curve find the point where your average (3.05) intersects [let's assume it is at 52%]. Now, when your average changes [to 2.77 or 3.33 in your example], you take that exact point (3.05, 52%), and slide the entire curve to the right or left so that 52% now intersects with 2.77 or 3.33.

This simple shifting of a bell curve should get you very close to where you need to be, and provide a better understanding of how scores are shifting. Now, just practice making accurate bell curves based on the distributions you have.

NORMDIST is what I used. Whether "normal" and disc golf should be in the same sentence is debatable :D